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Thanks, all!

wow, great info there champion, thanks!

Paulowich's Midgard Chess, Jumping Knights Chess, and Rose Chess XII has a piece called the War Machine, which leaps (0,1) or (0,3).

Ganymede Chess has a piece called the Frog, which leaps (0,1), (1,1), or (0.3). It also has a "Cardinal", not the conventional BN, but rather a Bishop+(0,1)+(0,3), as well as a "Fort", a Rook+(1,1)+(3,3).

Laugh, ok, here are some more.

'Operational Chess' by Andy Thomas, January 2006. 
Piece called 'General', leaps 3 squares away, all directions.
Thus, it is a 0,3 and 3,3 leaper, and also can move like the 3,1 Camel and the 3,2 Zebra.

'Scirocco' by Adrian King, 1999, piece called 'Frog'.
Leaps as a (0,3) or (3,3) leaper, plus can move 1 square all directions.

G.P.Jelliss gives these names and movements.
'Frog' {1,1}+{0,3}, 'Toad' {0,2}+{0,3} and 'Newt' {2,2}+{0,3}
Also 'Threeleaper' {0,3}, and 'Tripper' {3,3}.
The 'Threeleaper' and 'Tripper' are in my game 'Sky', under those names.
You can see the 'frog' {1,1}+{0,3} in 'Presiding Chess' by Tucker Kao, 2003.
In 'Quangtrung Chess' by Vu Q, 2002, piece 'Voi' moves like above 'Newt'.

Chess with Different Armies has the 'Half-Duck', steps 1 space diagonally, or jumps 2 or 3 orthogonally.

'The Travelers' by Roberto Lavieri, 2006.
Piece called 'Trey', has non-capture and capture moves, as follows.
Non-capturing: slides 1-3 squares orthogonally or diagonally.
Capturing: can leap 3 squares, 0,3 and 3,3, only to capture.

In 'Jetan', (Martian Chess) by Edgar Rice Burroughs, piece called 'Flier'.
The Flier moves three squares diagonally or in a combination of these diagonal directions. It may jump over intervening pieces.

I do not know of a piece that is a pure 0,3 and 3,3 leaper.
Not saying there isn't one, with so many chess variants and pieces, 
just saying i have not seen it, as far as i can recall.

Grin, this shows how little impact odd shatranj variants and "wargames"
make on even this group. I've used the pieces you've been talking about
in several games, but most tend to be obscure, at best. I would be
surprised if Christine Bagley-Jones has not used these in some game, and
Larry Smith might well have, also. 

Anyway, here's some games where various pieces go 1,2, and/or 3 squares
orthogonally or diagonally, and two where the R4A and B4D are used. ;-)
The
first one is a CwDA collaboration between Abdul-Rahman Sibahi and myself.
It uses the full 3-piece combo, the linear orthogonal 3 square mover, the
linear diagonal 3-square mover, and the combo queen-analog piece. Most of
the others do not use the combo piece, just the linear piece. In the
Warlord Games [warlord-2], the pieces leap to each of the 3 squares,
overpassing any intervening pieces. In the other games, the pieces are the
linear hero and shaman, moving 1, leaping 2, or doing both in either
order.
The two large multimove chieftain games use the R4A and B4D. 
http://www.chessvariants.org/index/msdisplay.php?itemid=MScwda:theshatra

http://www.chessvariants.org/index/msdisplay.php?itemid=MSchieftainchess

http://www.chessvariants.org/index/msdisplay.php?itemid=MSchesimals:auto

http://www.chessvariants.org/index/msdisplay.php?itemid=MStwolargemulti-

http://chessvariants.wikidot.com/warlord

http://chessvariants.wikidot.com/warlord-2

Right probably Grande Acedrez Lion is mis-interpreted, where some old schools describe distances including the starting square. Okay, Gilman defines Beaver as Bison plus Trilby in M&B06. Trilby is bi-compound of the Tripper (3,3) and Trebouchet(0,3) renaming the Jeliss and problemist Threeleaper, both Tripper and Trebuchet in M&B06. If Gilman has used any of those, they would be Tripper and Trebuchet, not Trilby and Beaver. He often deliberately implements a new-found or -named fundamental, but his indexes do not go from piece-type to cv anywhere reliably yet. Count on his having put all the oblique leapers into cvs, but where are Tripper and Trebouchet? Lavieri Grand Bishop of Altair leaps three diagonally as a bi-compound having to slide to the other Bishop squares. Heddon's Microorganism's Paramecium is Tripper but as just one part of the tri-compound it is. Heddon's IO's SuperComputer is Trebuchet but as just one part ot that tri-compound. Almost certainly Beaver and Trilby are not done yet formally in any cv. There would be several other piece-types that can leap to (0,3) or to (3,3). For example, Ramayana Buddha who leaps to all Rook destinations, and Ramayana Rakshasa who leaps to all the Bishop arrival squares. Jeremy Good's cv hre has both Archabbott and Squirrel as tri-compounds, the loophole for this larger topic.

(test)

Wikipedia and All the King's Men both name the (0,3)-leaper the "Threeleaper" and the (3,3)-leaper the "Tripper", but neither lists an inventor or a variant that uses them.

Betza's notation uses H for a (0,3) leap and G for a (3,3) leap, but I don't recall ever reading how he chose those letters.

Since you people are being so kind and helpful, I have another cluster of questions (my last for the foreseeable future).  Charles Gilman proposed the name Beaver (a bigger rodent than the Squirrel) for a piece leaping to any square on the third perimeter.  Has such a piece ever been used?  I am aware of the Camel, the Zebra, and the Bison; have pieces leaping to the third square orthogonally or diagonally or both ever been used?  I am aware of the Grande Acedrez, but as I noted there I think the lion has the dabbabah's move.

In response to Jeremy's points,

1) Of course the majority of the "space" will be empty for being generally unused pieces.  It would be nice to have an easy way to find out whether such a weird piece like the "forward-left knight, forward-right alfil, backward ferz/rook" actually was invented by someone and used in a variant (unlikely, for this one).  There is perhaps already a method for doing this, using funny notation or something similar, but a graphical interface would be nice.

2) For popular pieces, I'd be happy to have links to the already existing (perhaps large) articles here.

3) For pieces that move differently than they capture, the applet could also allow for distinguished move/capture flags.  For even more exotic movements (Mao, Moa, Cannons, etc.) you'd be out of luck (though for these examples the applet should probably just list them as possibilities when given the appropriate input of a knight or rook movement; the pages on CV could then clarify their distinctiveness).  I tend to think of en passant and castling as special rules to the game and not inherent to the pieces.  Further special abilities (multiple captures, custodial captures, copying piece moves, carrying other pieces) wouldn't be the focus of such an applet.

It's probably too much work to try to dig up as many pieces as possible to make this helpful, but I think it would be nice for something like the introductory comment in this thread.

Gilman's Man & Beasts system already names many tri-compounds and quadra-compounds specifically beyond the automatic tag each one gets by the fitting defined suffixes made into single word. Extensions of the M&B prefixes and suffixes are able to name in one word millions of piece-types made up of at least three component parts. Some would be contradictory and so inapplicable as Jeremy Lennert points out.  Some are discussed in the articles, for example ordinary Knight in 2d is HELM + HELMCOWARD + HELMDEXTER + HELMSINISTER in Chapter 12 http://www.chessvariants.org/index/msdisplay.php?itemid=MSmanandbeast12:. So that way of dividing things, even Knight is not fundamental, but a quadra-compound. How many tri-compounds appear as usefully making piece-types in Charles Gilman's 275 cvs? For follow-up also, what have been some other good tri-compounds in accepted cvs? Not over a hundred as unique different p-ts in 4000 or 5000 cvs.

Such an app would basically just be an interface to a giant table listing pieces for every possible combination of moves. Assuming you could create that table, the app would be straightforward. I question its usefulness, though.

Firstly, most of the table would either be empty or taken up with obscure pieces that few would be interested in. For every combination of squares you could light up that corresponds to a common compound, there's a ton of combinations like "forward and left as a knight, forward and right as an alfil, or backwards as a ferz or rook" that are just weird. I'm sure you can dig up names and variants for several pieces like that, but it could take a lot of work if you're really being thorough, and the odds of a user stumbling upon any particular one of them are pretty low.

Secondly, for popular pieces, there would be an unwieldy amount of information to display. Take a look at the Piececlopedia page for the Bishop-Knight Compound, for example; it's got a bunch of different names and has been used in a ton of variants (the list on that page can't possibly be complete, but it's already more than one screen of info).

Thirdly, your database would have to leave out (or misrepresent) a lot of important and interesting pieces, because pieces aren't defined only by which squares they move to. The Knight, the Mao, and the Moa all move to the same squares, but they take different paths to get there. Some pieces, like the Cannon, require a hurdle to move. Some pieces capture differently than they move; Ultima is a variant based around giving every piece a different method of capture. Some pieces can make special moves under specific circumstances, like the pawn's initial double-step or en passant capture in FIDE. Some pieces just have weird special rules, like the ability to capture multiple times during their movement, or copy another piece's movements, or move other pieces with them.

This exchange makes me wonder how hard it would be to create an applet that allows one to choose a small grid and fill in certain locations with flags, then outputs a list of named pieces (with links to appropriate CV pages) that share those moves on that grid (or perhaps--as options--at most those moves or at least those moves).

The Rook and Alfil piece is a very old piece, called the 'Ark'.
First mentioned in the 13th century 'Bonus Socium' manuscript, 1275.

Information from V. Nebotov’s Dictionary of Fairy Chess.

G.P.Jelliss talks about this piece in Variant Chess 2, April-June, 1990, showing the mate in 2 problem from the manuscript the Ark was in.

Rook Alfil Fers is in 'ABC Chess', by Jeff 'Cavebear' Stroud, 2001.
You create pieces and 'army2' compounds start with 'rook/alfil/fers' so that piece can be chosen.

Bishop Dabbaba is in 'Chess with Different Armies', called 'Bede'.

This makes me wonder: have the combinations rook+ferz+alfil, bishop+dabbabah, rook+alfil been invented and tried?