If you color rhombs with n colors at the four sides you get (n^4+n^2)/2 one-sided and (n^4+3n^2)/4 two-sided distinct pieces.

Colors One-sided Pieces Two-sided Pieces
Number Examples Number Examples
3 45 9x5 parallelogram 27 three 3x3 rhombs
4 136 17x8 parallelogram 76
5 325 25x13 parallelogram 175 seven 5x5 rhombs
6 666 37x18 parallelogram 351
7 1225 35x35 rhomb 637

3 colors, 45 one-sided pieces:

3 colors, 27 two-sided pieces:

Turning the rhombs and choosing matching colors for the borders we get a hexagon. Instead of colors we can also choose straight, male and femal borders.

4 colors, 136 one-sided pieces:

5 colors, 325 one-sided pieces:

The picture shows the method for the construction. Save all pieces with only green, teal, blue and yellow edges for the second part of the construction. A few pieces with one wrong color as the two pieces with one red edge are allowed, too.

5 colors, 175 two-sided pieces:

Since there are only five colors two couples of rhombs must get the same edge color.
6 colors, 666 one-sided pieces:

7 colors, 1225 one-sided pieces: