sorry to be late replying to this thread.
Polyominoes of order 3 do not exist I. N. Stewart and A. Wormstein
Journal of Combinatorial Theory, Series A Volume 61, Issue 1 , September 1992, Pages 130-136
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So you can't do it with a polyomino.
see also "Trisecting a rectangle", Samuel J. Maltby, Journal of Combinatorial Theory, Series A, v. 66, no. 1, April 1994, pp. 40-52. abstract: In this paper it is shown that it is impossible to dissect a rectangle into three congruent pieces unless those pieces are also rectangles. doi: http://dx.doi.org/10.1016/0097-3165(94)90049-3 math reviews: http://www.ams.org/mathscinet-getitem?mr=95g:52022 which seems to answer the original question asked. i note that it must be understood that the pieces are connected, for otherwise there are other (easy) dissections. this leaves open the case of dissection into 5, 7 or 9 congruent pieces. mike