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I would suggest in a large game with this few pieces that the Pawns attack
both diagonal and triagonal. The White Pawns only changing levels upward,
and the Black Pawns downward. Though I might allow both to make these level
changes while attacking to any available cell. Technically this allows the
Pawns to retreat, but since they advance a level it all evens out. ;-)

Also the Knights might be best if allowed the moves of the Super Knight of
3D Chess. That is the (0,1,2), (1,1,2) or (1,2,2) moves.

The Rook might also be allowed the planar movement. This would greatly
speed up the game, and provided a piece which effectively mates the
'slippery' 3D King. I would not give this type of move to any other
piece.

The Bishop might also be allowed the triagonal slide. The sames goes for
the Queen. Though with their diagonal slide, they attack their opposites
from the start. So there might need to be some form of restriction for
sliders when changing levels. How about this: they may only perform level
changing slides if friendly pieces are present on levels which they pass
through. This could effectively restrict their level change to one. And at
the start, they could only change two levels. This rule could also be
applied to the planar Rook.

And the twin pennies have hit the table. ;)