I've been thinking on this a lot lately and I think I've made the game
better.  I still have yet to program it into a computer or find someone
who is crazy enough to play, so it is all theoretical.

First off the diagonal move has changed.  That changes the bishop, king,
queen, and pawn.  Second I’ve redefined the knight.

Also as suggested I have come up with new names for some of the pieces.

The rook is now the octahedron, because it’s move makes the outline of an
octahedron.  The new bishop is a cube, because it’s move makes the
outline
of a cube.  The old bishop, not used, would be a cubeoctahedron.

(When I say 2d rook or bishop I mean the normal chess version as
translated into three dimensions.  So a 2d rook moves in a line forward,
backward, sideways, straight up, and straight down.)

The new bishop (cube) move is one moving as a 2d bishop, moving as a
three
dimensional mover (unicorn) or moving as a 2-d bishop then a
three-dimensional mover outward.  The new bishop is not colorbound.  A
picture of the spaces covered is here:
http://members.lycos.co.uk/christhecynic/hpbimg/Chess/cube.bmp

The pawn moves one step straight forward, or one step forward as a cube,
which is one step as an ordinary bishop or one step as a unicorn.
http://members.lycos.co.uk/christhecynic/hpbimg/Chess/pawn.bmp

The king moves as a bishop or as a rook, thus covering the entire 3x3x3
cube around it.  The queen moves as a bishop (cube) or a rook
(octahedron)
http://members.lycos.co.uk/christhecynic/hpbimg/Chess/king.bmp

The knight moves one step as a bishop then one step straight
(orthogonally) outward.  This no longer follows the “squares a queen
can’t
go to” rule, but it is closer to the appropriate value, though still
weak,
and it does seem a bit more elegant than anything else.
I wanted to show a picture of each of the pieces, especially the ones
that
were changed, but my computer goes to hell every time I try to upload a
new
image.

The rook (octahedron) as I think I’ve already stated moves as a 2d rook,
a
2d bishop, as a 2d rook then outward as a 2d bishop.
http://members.lycos.co.uk/christhecynic/hpbimg/Chess/Rook.bmp

The queen moves as a bishop (cube) or a rook (octahedron) I don’t know
what you would call it as cubeoctahedron would be the name of the old
bishop.
http://members.lycos.co.uk/christhecynic/hpbimg/Chess/RookCube.bmp
-

There may be space for additional rules.

For example a rook blocked on orthogonal adjacent squares could still
move
as a 2d bishop.  So perhaps it would be better to say that it moves once
as
a rook, then again at a right angle and then continues out as a bishop. 
While this sounds a bit confusing it should be simple to remember and
will
prevent the rook from moving when blocked.

The same can be said for the bishop, if all three diagonals surrounding a
3d move are blocked it shouldn’t be able to use that 3d move.

This is not likely to happen often, but if it does these two rules would
eliminate an annoyance.  On the other hand the rule itself might prove
more annoying to remember.

One thing that annoys me to no end is the freedom of movement behind the
pawns.  I can’t just add additional pieces to block it up, but I don’t
want the pieces there so free to move around.  One possibility, not
elegant at all but the best I can come up with, is to have the space
behind an unmoved pawn impassable.  The exception would be the starting
spaces of the major pieces.

This would definitely restrict movement, and it would be easy enough to
tell what was impassable, but it’s a bit of an odd rule I think.

-

I’ve also been thinking about other pieces.  I think it is best to
consider the Wazir as it is now, that being a one-dimensional single
step.
 That does something I consider odd though, it makes it so that a Wazir
rider is different from a rook (octahedron) also the Wazir, Dabbabah and
(0,3) leaper are all very, very weak compared to their bishop based
counterparts, which are not even colorbound.

The crooked bishop has two possible moves, one where it makes a line, and
one where it mirrors the rook (octahedron) I chose the second for the
reason I always thought of the crooked bishop as a bishop version or a
rook.
I wanted to show a picture of this, but as I said I can’t upload right
now.  It looks sort of like ripples on a pond, if that makes sense.

-

At some point I should probably ask them to updatye the page, but before
then I need to get the picture of the knight and probably make all of the
pictures smaller.