The 3D Bishop, Rook, and Queen each have 3 times as many forward directions as their 2D counterparts, but is the proportion more important, or is the absolute number more important?
The Rook has increased to 3 from 1, the Bishop to 6 from 2. Instead of having no forward forking power at all, the Rook now has a bit; instead of having the bare minimum, the Bishop now has very respectable forking power. It seems likely that the Bishop has gained more, but because the Rook has also gained one more sideways move than the Bishop (the Rook's extra gain is its ability to move straight up and down, for example from 1a1 to 8a1), so the ratio of value of Rook to Bishop is probably still nearly the same.
The Queen has 9 forward moves instead of 3; 9 forward moves sounds very frightening, and one suspects that the Queen is now worth more than Rook, Bishop, and Pawn.
In 3D Chess, the King has 3.25 times as many moves as he did in two dimensions; however, the board is 8 times larger, so the King's strength may actually be less than it was in 2D.
We know that the King's added mobility makes it harder to give checkmate. In the bare-king ending, it requires two Rooks and a bit of cleverness to force mate, while in 2D it needed only one Rook and it was easy. However, the added squares, the added number of directions, and the added mobility of the other pieces may make it easier to give check.
In some cases in the endgame, the ease of giving check will simply produce a boring draw; but in the middlegame and opening, the ease of check will produce more forks, and make it easier to accumulate a winning advantage.
Although each Rook has less value with respect to the King, there are so many Rooks that the ratio of King-strength to the combined total strength of all Rooks in 3D is much smaller than the same ratio in 2D. This should mean lots of mating attacks, and games should be shorter than you might otherwise expect: not 8 times as many moves as in 2D Chess, but only (for example) 7 times as many moves per average game. (That's still a long game, of course.)
From a random starting place on a board of size n, the probability that a square exists on the board at a displacement whose coordinates are (x,y,z) is given by "((n-x)*(n-y)*(n-z))/(n*n*n)". Multiply that by the number of directions, and you have the average mobility of a jumping piece. (For what it's worth.)
It is clear from this that the (1,1,2) jump only has seven-eights' as much mobility as the (0,1,2) jump, the (1,2,2) jump is even weaker, and the (1,1,1) move of the Bishop is worth a bit less than its flatland moves.
However, the (1,1,2) jump has a bit more forwardness, and is likely to be just as strong in practice as the (0,1,2) jump.
If this is really true, it probably increases the tactical complexity of the game to an extent such that one can no longer calculate, but must instead depend on shape, pattern, and intuition most of the time, and develop with experience a sense of when to try calculating sequences of moves. That would make 3D Chess a fine game indeed.
So far, this has been true in every case I have examined, except that the 3D HFD turned out to have the tactical property of winning huge material with its first move from the opening position. (Its general strength was probably okay, but in the specific position that starts every game it generated unstoppable threats.)