How many equilateral triangles have all three vertices among the eight vertices of a cube?

1. For a cube of edge $1$, two vertices are at distance $1$ (an edge), $\sqrt2$ (a face diagonal), or $\sqrt3$ (a space diagonal).
2. Three vertices pairwise at distance $1$ do not exist, and each vertex has only one vertex at distance $\sqrt3$, so the sides must all be $\sqrt2$.
3. Vertices pairwise at distance $\sqrt2$ are those of one of the two regular tetrahedra inscribed in the cube; each has $4$ triangular faces.
4. The number of triangles is $2\cdot4=\boxed{8}$.