I think you are asking, "Are there any gasoline engines which don’t have timing chains or timing belts, similar to diesel engines not have timing belts/chains?"
As far as I know, in today’s cars, all four-stroke gasoline engines have timing chains/belts, or gears which perform the same function. Two-stroke gasoline engines may be what you are thinking about. Maybe, a hundred years ago there may have been four-stroke gasoline engines with a different design; but, I’m not an automotive historian.
On the two-stroke engine, the piston itself acts as the intake and exhaust valve…as this little video shows: http://en.wikipedia.org/wiki/Two-stroke_engines