We were there June 19-26 two years ago, and it rained CONSTANTLY. It would be mostly cloudy in the a.m., with a few peeks of sun, and then come noon-time or so, it poured steadily into the evening. It would let up for a few minutes and then pour again. The locals said it was the most rain they had seen in 3 years! Figured it had to be on OUR trip!!

This went on every day we were there, except for the day we left (of course!) It was beautiful that day, all blue sky! LOL
We bought ponchos, and braved it with the best of them. There isn't much you can do about the weather, and if you let it stop you, you miss alot. We weren't HAPPY about it, but we made the best of it. The kids were disappointed that we didn't get to Blizzard Beach, but we did get to swim in the FW pool twice.
My best advice is to bring a couple of extra pair of shoes, and don't wear sneakers if it looks like rain! I saw alot of people wearing water shoes and rubber flip flops and that made perfect sense! LOL
BTW, I discovered ponchos at our local Dollar Tree...they come two in a package for $1. The ones at WDW are pricey and do rip easily!
Have a great trip!