I don't see why you'd need to increase the number of spaces on the board (other than obviously raising the power of
Remember, in a 3D chessboard there'd already be 8*8*8 = 512 cubes on the board - and half of these would be allocated to the players, 128 each.
Alternatively one could have four-player chess with four lots of 32 cubes in the "corners" of the board. Though this would obviously involve 3 rows of 8 pawns and the original 8 pieces, which wouldn't fit your extra piece per dimension. Also, pawns would need to be able to move in multiple directions.
If we stick with my first idea, one possibility is to only have the center 16 cubes used for non-pawns, with the outer 48 cubes of the non-pawn layer empty. This is exactly double the number of non-pawn pieces of a 2D chessboard, which could be in the following quantities:
4 rooks (move out of the 6 faces of a cube)
4 knights (move 2 cubes in one direction, one cube in all other directions)
4 bishops (move out of the 8 vertices of a cube)
2 of your new piece type (move out of the 12 edges of a cube)
The only problem here is that in order to preserve symmetry the 4 knights would all have to go on one color of square and the 4 bishops on the other.
Alternatively if we really do have 64 non-pawn pieces (which does seem overkill), they could be like this:
16 rooks (at positions 0,0, 0,1 and 2,2, reflected around the layer in full symmetry)
16 knights (at positions 0,2 and 2,3, reflected around the layer in full symmetry)
16 bishops (at positions 0,3 and 1,2, reflected around the layer in full symmetry)
12 of your new piece type (at positions 1,1 and 1,3, reflected around the layer in full symmetry)
3 queens (they would probably have to have a different name !)
The only remaining problem is that in either situation there would be at least two of some pieces which both start on the same color cube (light or dark) so you couldn't refer to them as "the light bishop" or "the dark rook" etc.