And while hexagonal boards may be in scope for the ID, I imagine 3D and hyperbolic boards are far from it ;)
Representation of 3D and 4D 'boards' is mostly problematic for the human player. One often resorts to displaying 2D slices of the board next to each other, which basically maps it to a larger 2D board. Chess programs in fact use the very same technique to map 2D boards to their 1D memory, storing them row by row. In all these cases separators between the slices would be needed to prevent 'wrapping' from one slice to the other. Usually this is done by separating the slices by enough inaccessible cells of the 2D representation that the leaper with the longest range cannot jump over it.
E.g. for a 5x5x5 variant with range-2 leapers ('Knights') you would map it to a 33x5 (or 5x33) board, with five 5x5 playing areas separated by four 2x5 'guard bands'. On this board an orthogonal step perpendicular to the slices would be a (7,0) leap, and is representable in XBetza as WXX. So the Raumschach Rook would be RWXX4, a Raumschach Bishop BHX4DXX4FXX4.
Representation of 3D and 4D 'boards' is mostly problematic for the human player. One often resorts to displaying 2D slices of the board next to each other, which basically maps it to a larger 2D board. Chess programs in fact use the very same technique to map 2D boards to their 1D memory, storing them row by row. In all these cases separators between the slices would be needed to prevent 'wrapping' from one slice to the other. Usually this is done by separating the slices by enough inaccessible cells of the 2D representation that the leaper with the longest range cannot jump over it.
E.g. for a 5x5x5 variant with range-2 leapers ('Knights') you would map it to a 33x5 (or 5x33) board, with five 5x5 playing areas separated by four 2x5 'guard bands'. On this board an orthogonal step perpendicular to the slices would be a (7,0) leap, and is representable in XBetza as WXX. So the Raumschach Rook would be RWXX4, a Raumschach Bishop BHX4DXX4FXX4.