jmc

The distances between corners of a unit cube are 1, $\sqrt{2}$, and $\sqrt{3}$. If the two shortest sides are 1 and 1, the third side has to be $\sqrt{2}$. If the two shortest sides are 1 and $\sqrt{2}$, the third side must be $\sqrt{3}$. If the two shortest sides are $\sqrt{2}$ and $\sqrt{2}$, the third side must also be $\sqrt{2}$. There is no case where the shortest side is $\sqrt{3}$. This gives $\boxed{3}$ non-congruent triangles.