Ok. It's easy to see that rectangle cannot be cut into odd number of equal triangles: Monsky proved [here] that a square cannot be cut into an odd number of triangles with equal areas. But if we have rectangle divided into odd number of equal triangles, we could shrink it in one direction and get a square divided into an odd number of triangles with equal areas - contradiction.
Update: If the tile could be cut into odd number of triangles with the same area, then obvious that rectangle cannot be cut into odd number of such tiles. Example:
