I think scan-line fill algorithm is used in this case. Its idea is described on the image bellow:
The problem is that he is not rasterizing a 2D projection of a 3D triangle, but rasterizing a 3D triangle in 3D voxel space.
This is REALLY bugging me! No matter the orientation of the triangle in 3 space, it is still a planar 2D object. This *shouldn't* be hard to figure out and I'm stumped!