Here is my idea for a much cleaner (or at least much easier to understand) cycle which may be an improvement.
There are four piston chambers A, B, C, and D. Gas flows in the direction A->B->C->D->A, and the corresponding chambers which gas flows between have connections with valves controlling whether gas can flow though each of them. The pistons are opposing, so A and C expand while B and D contract, and vice versa. There's a heat source continuously warming up A, and a sink continuously cooling down C. All pistons are the same length, but A and C have the same diameter while B has a larger one and D has a smaller one.
The cycle is in four phases. They're all done by a single reciprocating piston motion, but I'll explain how one of the two unconnected regions of gas behaves to make things clearer.
First, chamber A is fully expanded and B is contracted. The valves between the pair A and D and the pair B and C are shut, while the one between A and B (and also C and D) are open. Heat from the source causes expansion of the gas, which forces B to expand and A to contract (remember that B has greater diameter than A and that the piston motions of the two are directly connected). When B is completely expanded and A completely shut, the valves are flipped, so that the valve between A and B, as well as C and D are shut, and the valve between B and C (and D and A) are open. The cooling in C will then cause the piston to go the other way, emptying B and filing C as the overall volume goes down. Then the valves are flipped again and the gas flows from C to the larger D due to further cooling, then the valves are flipped one last time and the gas flows from D to A as heating expands the gas, and then the cycle repeats.
In a complete system there will be two cycles going at all times in two mostly unconnected regions of gas, but leaking between the two regions is no big deal. The two going at once has the nice property that the heating chamber is expanding at all times and the cooling chamber is compressing at all times.
This approach has the added complexity of having actuated valves, but that's not such a big deal. Internal combustion engines rely on valves and they work just fine.
This mechanism appears too simple to not have been thought of before. So my question to everyone is: Are stirling engines just more thermodynamically efficient than they appear to my uneducated eye, or are there some losses in my proposed system I'm unaware of, or has this mechanism been thought of before and simply not used because of its greater mechanical complexity? Or is this actually a possibly useful innovation?
Update: My intuitions about thermodynamics were a bit off. This design will work, but heating during expansion and cooling during compression turn out to be no-no's from an efficiency standpoint.