Well, since Dave is out of town, I'll throw in my .02. Late June early July is probably your best bet, if you are wanting to keep the bugs to a minimum. Even then, the skeets are going to be bad as are the no-seeums, etc... Problem is the dang things come out as the snow melts. Its either cold, buggy or raining. Sometimes all three. Take your pick. If you are going to the interior for gold panning (quite a bit is done in Southcentral and the bugs are less, but so is the gold) early to mid June is the time.
If you decide to come up in July, just bring your headnet, plenty of bug dope and rain gear. And have fun.
A good friend of mine goes up year to do some gold panning. Due to the work he does, he usually goes in July or August. Don't ask me to tell you where he goes, because he won't tell me. He's brought back quite a bit. Mostly for his collection, not to make money on.