Our ground was far too rocky and hard to even consider digging under and placing hardware cloth. So what we did was run hardware cloth up about 2 feet as a skirt on the lower portion of our run, then we folded it outward at the bottom and ran it about another 2 feet out. We pinned it down with landscape fabric staples. We did plan to cover it with rock, but we procrastinated. Lo and behold, the grass grew right through it and you can't even see it! But it sure does the job!
Our English Setter, Molly, who was very accustomed to the chicks being kept where they were, went nuts when they were turned out into the run for the first time. She paced the full length of the run over and over again, then tried to dig under it. She came away with a broken and bloody toenail for her trouble. She shot the chickens a dirty look over her shoulder as she limped over for a little sympathy and never bothered them again, even when they were free in the yard. She must have blamed then for her discomfort. This is the system we've had in place for a couple of years now, and it hasn't failed us yet.
As for what to put in the run, we used pine shavings for a long time, but found that they just don't break down like we thought they would. So now there's some pine shavings, but mostly whatever we have on hand - garden trimmings, dry leaves (we bag them during the fall and set them aside to add periodically) grass clippings, a bit of straw now and again.....whatever is handy. In the summer our run is covered with landscape fabric for shade because air will circulate right through it. And it does a pretty good job of repelling rain as well. You wouldn't think so with it being permeable, but if you've ever put that stuff down around plants and then watered it, you've probably noticed that the water puddles on it, runs down along any creases until it finds the opening where your plant is, and any that stays on the surface of the fabric takes its sweet time soaking through. With the fabric shaped in an arch like it is on our run, it simply runs off. We used metal lath with predrilled holes, attached the fabric to it with twist ties, and we can roll it up and down like window shades!
Landscape fabric "window shades". The lattice we just added because our setup is visible from 2 sides of the street and we live in town. Didn't want to have an eyesore in the yard.
If you look carefully you can see the hardware cloth skirt going partway up the run, then extended out at the ground.