Believe it or not, sand. Lots and lots of sand. We, too, have clay (to the point I can throw pots with it, lol), so when the mister starts improving a patch to turn into more garden (the goal, as is yours, is to have no lawn left), he adds sand, compost (we make our own), peat moss, and vermiculite. He adds other stuffs too, but he's the gardener, not me, so I don't know all the bits.