I believe the problem of dissecting a square into an odd number k of congruent pieces has been studied fairly heavily. Last I knew, for k <= 15, the only known solutions were with rectangular pieces in a grid arrangement. To my knowledge, no proof that this must be the case exists for 3 <= k <= 13. For k = 15, a solution with non-rectangular pieces is obtained by tiling a 9 x 5 rectangle with 15 L-triominioes then stretching the rectangle into square shape.