Update on this: I used the milk jug idea someone posted in another thread and was quite impressed - I had several dozen dead flies when I came home that day.
And then I brought home the Rescue Fly trap, that you add water too. It does stink like a dumpster, but I don't smell it once I'm 10 feet away or so, so nothing that will bother my neighbors. And wow! I've had it up for a day and a half and I've got to have at least 500 flies in there, and at no point do I walk past it and not see 5 more flies trying to get in. I've been moving it all around the yard and it just draws the flies to where ever I put it. I can't get another one until the weekend so I hope it can contain all the flies I catch the rest of the week.
It's a lot better out in the yard, the remaining flies mostly hang out around the trap, and I only have a few buzzing around the chickens at any point.
I think my plan of attack is going to be to get a couple more of these next weekend and also order the fly predators. Then hopefully I'll be getting the existing adults with the trap and getting their babies with the predators.
I am hopeful!