What's so Great about Great Chess?
Although I've never had the occasion to actually play any form of Great Chess, I've avidly studied the rules of many. It seems to me that there are several things that distinguish Great Chess from normal chess; furthermore, some of these things are already true of 8x8x8 3D Chess.Games of Great Chess normally have larger-than-normal boards, and the 8x8x8 board is bigger than any of them. Great Chess always has lots of pieces, and 8x8x8 has even more.
Games of Great Chess usually have more different kinds of pieces than there are in 8x8x8, and there are usually many fairly weak pieces to go along with a few strong pieces. I think there's a reason for this: the large number of different pieces serves to make it harder to coordinate your army, and the weak pieces make the game less tactically intense (but perhaps strategically deeper).
Games of Great Chess usually have a larger proportion of empty space on the board. This makes things a bit less tense and gives more breathing room.
Using the above ideas, it should be possible to design an 8x8x8 3D Great Chess.
P.S. according to the above ideas, Xiang Qi is a form of Great Chess.
8x8x8 3D Great Chess
First of all, take away the Pawns' initial doube-step. This makes the board a bit longer in effect, because the pawns need a bit more time to make contact.Next, empty the outer rim of the board. Now each side has only 36 Pawns and 36 other pieces, so there's a bit more breathing room. (This also means that Castling is no longer part of the game; and with all the edge Pawns gone, the King wouldn't find as much safety there in any case.)
Now let's have a White K on 4e1, Q 4d1, RN (Chancellor) 5d1, and NB (Archbishop) 5e1.
Put White Rooks on 2b1, 2g1, 6b1, 6g1. If 1. R2b1-2a1, R2b8-1b8 is a fair reply.
Griffon 3b1 and 5g1. Black's Griffon 3b8 opposes 3b1. This piece makes a one-square diagonal move and then a Rook move away from its startingpoint, and is historically a GreatChess piece. In its 3D version, if the diagonal move rises, the Rook also rises, and so on. Example: Griffon from 3b1 to 4c3 and if empty may continue either 5c4-6c5-7c6-8c7 or 5d3-6e3-7f3-8g3
Angriff (Anti-griffon) 6c1 and 2f1. A one-square Rook move and then a bishop move; no doubt this piece already had a name, but I made up a new one because I couldn't remember the right one. Example: Angriff from 6c1 to 6d1 and then 6e2-6f3-6g4-6h5.
Throw in four Knights and Bishops, and now there are only 16 empty spaces left, enough for 4 each of 4 different pieces.
We also need them to be weak pieces, so it's pretty easy...
The Snark must of course be one of the pieces, and the others might as well be a Dabaaba, an Alfil and a Commoner.