At my last house I dug out about 6-8 inches all around, put down hardware cloth (I'd use vinyl-coated as it will last longer) and then cement blocks for the coop and run boards to sit on. The hardware cloth came up and over the cement blocks and the vinyl coated steel wire I used to cover the run and area below the coop. Then I put hardware cloth around the run flat on the ground, two feet out, with rock on top of that maybe a foot out from the coop.
This time, my coop/with attached small run (whole thing is 5'x10', I think) is on 6"x6"posts, I dug out 6" so that the posts are flush with the ground, and I put hardware cloth in first so it wraps up around the posts. I have an additional run with a chicken door from the small run so I can let them out into the larger run when I want. It's 6'x30', and runs the length of the garden. I just have hardware cloth flat on the ground 2' out from that, with rock on it once again.
Since there is 6"-8" of dirt on top of the hardware cloth, they can dust bath and scratch, no problem.
I've had fox around my run a fair amount, but so far, hasn't figured out a way in (I'm knocking on wood now.)
Anything you can do to protect the birds is worth doing.