I know from personal experience that the Cooper's hawks will try for chickens; however, the only one I've seen try for ours was a juvenile, and he eventually gave up. The roosters would see him and give a yell, I'd come bolting out, the hawk would leave . . . I gave him a bad scare twice when he was bouncing around trying to grab a hen hiding underneath a tree branch on the ground, and I think having this lunatic human charging after him shouting really put him off his lunch.
When one of our trees drops a branch, I lug it into the chicken yard and set it up so that it creates a hiding place for the chickens. Having a place to hide from aerial attack has saved some of our chickens. They also spend a lot of time hanging out underneath a mulberry tree; I think they know that the hawks are out there. Our roos are very noisy about hawks, although I have yet to see one go after a hawk.
The string-grid idea is a great one, and if we start having more hawk issues I will use grids to create no-fly zones for the hawks. The hens range in a very large yard that would be nearly impossible to roof, even with a grid, but it would be much easier to create hide zones for them all around the yard.
Knowing what kind of hawk you're dealing with makes a big difference. Coopies are bird-eaters, so they're a risk; they're also nimble as all heck in the air because they hunt by chasing birds through tree branches. I have had a Coopie get into a chicken run that was entirely walled and roofed with chicken wire, and to this day I don't know how the little bugger did it. (When he realized that he was trapped in there, though, he was so upset that he didn't get any birds. We had a perfectly lovely time peeling his hysterical self off the wire that night and releasing him.) Red-shouldered hawks seem less interested in chickens, although all bets are off with chicks and bantams: a red-shoulder is pretty small and would have a heck of a time carrying off a large fowl, but might try for something it could carry. Red-shoulders are also territorial, and will gang up on redtails. Red-tailed hawks are big enough to be a potential threat to chickens. Great-horned owls are a real bear; I've had them try for our peafowl, but they bounced right off the Top-Flight netting. <evil grin>
In our yard, I have seen red-shoulders, redtails, Coopies, American kestrels, a sharp-shinned hawk, white-tailed kites, a prairie falcon, great-horns, and barn owls. Of these, only the Coopie has tried for the chickens, and he was a juvenile. There's plenty of local prey that aren't chickens, and the adult hawks around here seem to know better than to mess with them. It's just the juvies that are hungry enough, and inexperienced enough, to make a try for it.