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Okay, let's see if I can weave the 2 parts of my last comment together
here. Was discussing higher D diagonal pieces, and comparing them to alfils
because they may have rather limited movement. Another comparison is to the
lance, the 1D rook, a piece that can strike deep into the heart of the
enemy position, but has limited lines of action. 

But neither comparison is really valid, because the fully higher D diagonal
piece, once it gets into a position, can devastate it. Since such a piece
has far more degrees of freedom to move, a fully 3D diagonal slider can
easily fork half a dozen other pieces located all over the board. You have
to tame the pieces.

The traditional way to control the sliders is to clog the board with pawns.
Chess starts with the pieces hemmed in by two pawn walls. 8x8x8 3D uses a
pawn plane for the same purpose, 64 pawns occupying an entire 2D 'board'
in front of the 64 pieces in the game - for each player. 128 pawns can clog
up a 3D board pretty well, but this game must be a nightmare to play. 

This idea seemed so much more compact in my imagination; seems to be
sprawling out here, though, much like that 3D game in the last paragraph.