Only just got around to listening to this[1]. I haven't seen the original 'request' for a hex grid, supposedly to help with diagonals, but you need to consider that it
may give you two sets of diagonals face-to-face, but
either sideways or up/down (your choice[2]) are now corner-to-corner passes. Which is at least as annoying if you aren't happy with that from the start.
Coordinating hexes is a
little awkward. You can do a skewed grid[3] of /_/ or \_\ 'squares' or alternate miss-odd/miss-even[4] without skewing (assuming you don't use 'halves', or don't explicitly but assume[5] so). Yeah, I'd say that if you're not (re)starting out anew, best avoid trying to convert.
(Also, the standard cube is a hexahedron, as I'm sure you realised later. Though you mean something with >6 edges with non-orthagonal face-to-face angles, which is a bit awkward. You probably should take a cue from the (not necessarily platonic) solid that uses the 'kissing' points of
one or other packing of spheres. I think an edge-kissing lattice of octahedra (fudge the unkissing gaps between, or overlaps if you nudge them closer to have face-centre touching face-centre) as is probably most logical but might be
all diagonals[6], or else choose just 1D out of the 3 to be axis-orthagonal if you twist it accordingly.)
I think I had that particular hyperbolic game on my prior tablet. I must get it onto this one some time. If you could tightly pack icosohedra in (necessarily) non-Euclidean 3D space then you'd have
plenty of player-choices for straight lines - but weirdness in general and very few practical right-angles to any given axis!
Hah, I'm rambling. Don't mind me, the thing next in my listening queue that is playing to me now (no further D-things, nor E-things or F-things, it's a G-thing!) is distracting me from properly composing this. So E&OE, possibly. Someone else'll spot an elementary geometric error that crept in, I'm sure
[1] I have a lot of things 'queued' and I try to go through in wrappedharound alphabetic. Just got (back) to the Ds...
[2] ...or skew the hexes so that
neither orthagonal is represented, say 45°, 105°, 165°, 225°, 285°, 345° and wrap.
[3]
2,4 3,4 4,4 5,4 6,4
2,3 3,3 4,3 5,3
1,2 2,2 3,2 4,2 5,2
1,1 2,1 3,1 4,1
0,0 1,0 2,0 3,0 4,0
[4]
0,4 2,4 4,4 6,4 8,4
1,3 3,3 5,3 7,3
0,2 2,2 4,2 6,2 8,2
1,1 3,1 5,1 7,1
0,0 2,0 4,0 6,0 8,0
[5]
0,4 1,4 2,4 3,4 4,4
0,3 1,3 2,3 3,3
0,2 1,2 2,2 3,2 4,2
0,1 1,1 2,1 3,1
0,0 1,0 2,0 3,0 4,0
[6] Unless you prioritise corner-to-corner travel, when its two
mutually exclusive but entwined cubic packings.