Imagine doing the other sides of the cubes. Now orthogonal layers will be intersecting. There's no correct order in which to draw intersecting polygons. You either need to split them (and then you're back to where you started), or you need Z-buffering.
You _can_ actually do this with just one quad per plane (or maybe 2 quads - one for the front face and one for the back face)... although you're right that you can't do it using CSS (yet).
A few years back, I wrote a tech-demo^W game for Ludum Dare (http://ludumdare.com/) based on using GLSL shaders to ray cast within a cube containing voxels. I then used these cubes as 3D "sprites" for the game.
