I love running shoes for bike riding, personally, unless you're talking about minimalist running shoes, then no way.

What kinda flat pedals are we talking about?

If it's something like this, there are much better platform pedals.



I love Odyssey Twisted PC pedals. Nice, big, concave surface, pins are good at keeping shoes in place. Nice price, too. Can't guarantee they'll help the sore balls of feets problem, though.