We have wire over the top of the run, but it has a cover over the end next to the coop. That gives the girls a nice dry place to hang out if it's raining hard. We also don't need to shovel that end, just so they can get out of the coop in winter.
That's the area that ended up as dirt, since I don't water it, either. We have a lot of clay here. When the big snow drifts melted last spring, it got really muddy, so we put sand in. It really made it nice.
The girls have a big dustbath out in the yard, with sand, some fine dirt and a little DE. Their run dustbath is one of those large black rubber bowls, that we originally used for our sheep. When they aren't dusting or lounging in it, they like to perch on the edge. I think it's comfy for their feet. It's filled with mostly sand, although we also add some DE. They also like to just scratch around in it, like in the rest of the sand. When it's rainy, the yard dustbath gets too wet, but the run dustbath is under cover, so stays dry for them.