I buy the sticky fly traps at TSC or walmart. They're the kind that come in a little container and you pull and a long strand of sticky ribbon comes out--they come with a little tack too--you tack it up in your barn and voila!
I'm thinking flies are a good excuse to get some anoles from the pet store!!! and have a cage in the barn (away from the chickens of course) I love those little lizards.
I use mothballs...Has worked great to keep unwated bugs away!
ETA: Mothballs are VERY toxic so be sure to make some kind of container like I have to place the mothballs in a put small holes that nothing can get to them but will let out the smell of the mothballs.
Get some fly predators. You can find them online. Properly used they will keep those flies down to a tolerable level. they don't affect the adult flies, but they attack the pupae.
Welcome to BYC, Puget Sounder! I'm in Tacoma, too. I live in S. Tacoma & have 10 chickens (3 adults, 2 tweens, 5 chicks).
Flies are a constant problem for me, too. I'm headed to Lowe's tonight or tomorrow to get some fly paper. Last summer I created homemade fly traps (google for how-to) and did catch a good number. I found that the best fly bait was a nice, fresh, stinky pile of cecal poo (the gooey chicken poo that's every 5-6 turds). They did work to an extent, but not as much as I would have liked. Since I'm on a small lot, I really want to be contientious of my neighbors who are closest to the coop, since they'll get the brunt of the flies.
I sprinkled diatomaceous earth around my yard on Sunday, only to have it rain yesterday.