Alice's Chess in 3D? Not for me, anyway; but it's well-defined.
Simple games such as Cylindrical Chess, Billiards Chess, Chessgi, Knight Relay, Progressive, Avalanche, Dynamo, Grid, and so on, are all possible and enjoyable, and are well-defined as they stand. Others fall into this category; the ones listed are those on which I have spent a few moments of thought.
Momentum Chess in 3D seems well-defined but too confusing.
Games of the co-chess family would need some adjustment of their rules to be playable.
Trapdoor Chess in 3 dimensions is an interesting idea; likewise Conveyer Belt Chess, Rotating Grid, Pinwheel Chess, and other variants with moving boards. The rules would need some work, to redefine the way that the board moves.
Liar's Chess was an attempt to create a form of Kriegspiel without a moderator; limited lying in the 3D game would be interesting: you can only have fibbed about which level you went to.
Newton's Chess is my name for the game that I have discussed as a form of Momentum Chess, and which has come up in some email discussions of Billiards Chess: the pieces have different weights and speeds, and their continuing motions are defined by Newtonian mechanics. In 3D Chess, you could include gravity.
(Nobody has been able to make a proper game out of Newton's Chess because of the problem of fractional speeds or positions -- if a Pawn bumps a Rook, the Rook gets a speed of 0.2 squares per turn. Nobody wants a game with fractional speeds.)
Berolina Pawns are simple and interesting.
Foremost among these must of course be Progressive Chess.
Progressive 8x8x8 3D Chess. Too tactical for human beings to play?
This was done "by default", as the simplest sort of setup to start with; you can use 8 normal chess sets and simply put a crown on the one royal King. However, it is not a bad setup because the Commoner is an especially good piece to have when you are in the late endgame and are trying to mate the lonely enemy King with your few remaining pieces.
However, nobody with chess variants in their blood can look at that setup without wanting to put in some Chancellors and Archbishops and Knightriders and so on....
Radial symmetry is another idea: you have 4 Rooks in the corners 1a1 8a1 1h1 and 8h1, and.... Too many kinds of pieces.
You'd have to design a whole new game for this, of course.
I suspect that a 9x9 main level will work out better; with 32 pieces and 33 Pawns for each side, you'd have only 3 empty ranks between the armies, or with 9 pieces and 23 Pawns you'd have 5 ranks between. The "Pawns" should be several different types of Pawn-strength pieces, and because of the shape of the board, Berolina Pawns seem natural.
Each player has half a board. Before playing any move, you push two half-boards together to make a whole board. When it is your turn to move, you get to move once on each half-board where you have a piece.
In 2D, with only two players this is exactly the same game as FIDE Chess.
In 3D, this could mean that you attach to the other board at any angle to make your move, thus quadrupling the power of any move that threatens to cross the midline.