I wanted to jump in here and say I've been using straw in the run for 4 years. My run is 4000 sf and therefore not covered at all, except with bird netting. I do have covered areas that stay dry. I have never had the straw get moldy. It just eventually turns to dirt. And the chickens love straw day. You might say they have a hay-day digging through it and helping me spread it around. We live in a rain forest and moss grows everywhere, but I have not had mold in my straw even one time. I use straw in the coops and scoop that out into the run every week too. Last week I had puddles everywhere out there. Friday I spread a few bales of straw around and it's so much better.
I vote straw.