Yes, I think I have a good suggestion for you: Have you seen PowerGrip pedal straps? It's a rubberized-cloth strap that runs diagonally across a non-clipless pedal. I've been using these for years with Converse Chuck Taylor low-topped sneaks. It works great for me.
First off, the PowerGrip straps are really unobtrusive. Because of the diagonal arrangement, you slip your shoe in (the rubberized stuff makes the straps stiff enough to stand up off your pedal), and as you align your foot, rotating your heel in toward the bike, it tightens the strap's grip just right.
Getting out of these PowerGrips is really easy, because as you rotate your heels out, they get loose and your foot comes right out. (I have vintage clips and straps on an old road-bike, and the PowerGrips are so much better.)
Chuck Taylor's are a bit flexible, but the soles are fairly thin and non-spongy (which is the other problem with running shoes). Furthermore, the toe area is not bulbous in width, which works better with the PowerGrip straps.