Golomb's book Polyominoes has a section on this. Call the smallest odd number of copies of a polyomino that can tile a rectangle its "odd-order". Then Golomb says there are polyominoes of odd order 1, 11, and 15+6t for all $t \ge 0$. The polyomino of odd order 11 is due to Klarner , and is illustrated here by Michael Reid.
Reid has lots of pictures of tilings of rectangles with polyominoes. In particular the 15+6t family can be seen: here are polyominoes with odd-order 15, odd-order 21, odd-order 27, and so on. Reid has shown  that other odd orders exist, including 35, 49, and 221, but I don't know if there's a general pattern.
Finally, Stewart and Wormstein  proved that polyominoes of order 3 do not exist. (Stewart's book Another Fine Math You've got Me Into suggests that Wormstein is a fictional character.)
 David A. Klarner, Packing a rectangle with congruent N-ominoes, J. Combin. Theory 7 (1969) 107-115,
 Stewart, Ian N. and Wormstein, Albert. Polyominoes of order 3 do not exist. J. Combin. Theory Series A 61 (1992) 130-136.
 Michael Reid. Tiling Rectangles and Half Strips with Congruent Polyominoes. J. Combin. Theory Series A 80 (1997) 106-123.