Any kind of straw will be fine to use in the run. Do a thread search for deep litter management. Many of us keep our runs covered with deep litter. It essentially creates a nice rich spongy compost that: attracts beneficial insects, worms, bacteria and fungi to help balance the soil, helps to keep pathogens in check, improve the gut flora of your birds, thus improving their digestion and immune system. Gives them a job to do as they go to work churning that material into compost while they search out the insects and other goodies. It also keeps your run from turning into a mud pit or dust bowl.
The one warning I give against using straw, or even commercially purchased hay is this: Many farms are using some highly toxic herbicides in their fields/crops that can persist in the straw/hay for a very long time. In the case of hay: these newer herbicides are taken up by the hay, pass through the stomach of the cow and into the manure. The manure is used for compost or in the garden of an unsuspecting gardener,, who then can't figure out why his tomato leaves are stunted, curled and yellow. These herbicides can ruin an otherwise productive garden for years. KNow where your materials are coming from and how they have been handled.