This polyhedron can be made from 4 colors. Just like a cube there will be 3 colors for your square faces and same-colored square faces will be opposite from each other in the polyhedron and in parallel planes. Your triangular faces can all be the same color.
You will need to prepare 6 square faces in 3 of your colors (2 squares of each color) and 8 triangular faces (all of the same color).
To see the full-sized template click on this image:
To assemble this polyhedron you can surround one of the triangular faces with 3 squares of different colors. Then surround each square with 4 triangles. Place your next square face exactly opposite from the square face of the same color. Square faces of the same color will be, as usual, in parallel planes. Around every triangle there will be 3 squares.