Aha! Thank you, Mike, for successfully searching the literature!

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.

Shh, don't tell anyone, and I'll put a copy at http://people.brandeis.edu/~kleber/priv/trisection.pdf The authors begin by saying that they are actually generalizing the earlier result which is precisely our original question about the square. The congruent square trisection problem was posed in Crux Mathematicorum in 1983 by Stanley Robinowitz of DEC, and solved in 1991 by Sam Maltby, then a student at U. of Calgary. The solution appears in Vol 17, and by happy chance, that's the old volume of Crux that's available free on their web site, at http://journals.cms.math.ca/CRUX/v17/ In particular, p.141, http://journals.cms.math.ca/cgi-bin/vault/public/view/CRUXv17n5/body/HTML/14... is the first of six pages of the result. I haven't read the entire proof, but the line of attack does indeed start roughly the same way our discussion here did, and in particular it is very 3-specific, and not at all amenable to generalization to the 5-piece case (which Gardner claimed was true at least 30 years earlier!). --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.