If the smelly stuff is getting into the fabric on top, then this probably means washing them every time you run, which would obviously be a problem, as they take a while to dry (BTW - I put a pair of trainers in the hot air dry cycle of my washing machine - and the glue melted and I ended up with a pair of completely knackered new £70 shoes).
To get the mud etc off the underside and edges, I put an inch or so of water in a bowl, and stick the shoes in there, using an old toothbrush to get off the stubborn bits and bits around the edges. You can then tip the dirty water down the sink.
HTH, Iain