Packing four-sided dice

A while back I was trying to work out the geometry of a game map with a triangular grid, instead of the typical hexagon. Somewhere along the way I got sidetracked by the idea that if I could figure out how to subdivide 3-D space into tetrahedral blocks, this would be the basis for games with a simple sort of 3-dimensional movement. I got busy with my pencil and sketched and scribbled for a while, trying to see if I could work it out, but I kept getting stuck. Realizing that this was just the sort of problem that better mathematicians than I are likely to have worked out already, I did a bit of searching. It turns out I was wrong:

Packing Tetrahedrons, and Closing In on a Perfect Fit

Aristotle mistakenly thought that identical regular tetrahedrons packed together perfectly, as identical cubes do, leaving no gaps in between and filling 100 percent of the available space. They do not, and 1,800 years passed before someone pointed out that he was wrong. Even after that, the packing of tetrahedrons garnered little interest. More centuries passed.

I was wrong - you can't do the sort of subdivision of 3-D space that I naively hope for, but at least I'm in pretty good company there.

Here's another source for the same story: Packing Tetrahedrons, and Closing In on a Perfect Fit
And a relevant Wikipedia article: Tetrahedron packing

If I switch to octahedral blocks, my original idea might still work. Maybe I'll buy a bunch of 9  8-sided dice and test it the easy way.

[Edit: Nine-sided dice ... that's a good one! ;-)]