Here's a picture of part of our run (it's not finished in this picture but gives you an idea):
It's a lot like Wynette's except that we buried our 1" chicken wire almost 18" into the ground. We also used cement with our 4x4 posts and spaced them 8 and 1/2 feet apart. Our run is six feet high so we can walk into it, I don't have it covered at this point as I have goats in with my chickens and the hawks have thus stayed away. (Not to mention the dogs and cats "guarding" the place. /img/smilies/big_smile.png
I am making a separate run for my chicks that will be covered since hawks are far more likely to go for them than the adult hens and the goats won't be in with them.
I used 2x4 welded wire only because I can't afford hardware cloth. I did use hardware cloth over all windows into the coop.
This second picture shows what we did under our run gate: that will likely be your weakest link so this was our solution. It's worked great -the gate doesn't get caught on anything and nothing can dig under it as we poured the cement about 18" deep also.