We went June 22-29th of 2008. It was hot, but no worse than home. It was humid, again no worse than a home. I think that if you live somewhere that has nice summer temperatures, then you might be bothered by the heat. It was hot there and it was hot at home.
Crowds: We went knowing that it was going to be crowded so we made plans to be be at the parks early, avoid EMH parks and use fast passes. We really did not have a problem with the crowds. proabably the worst crowds we experienced were during the parades.
Weather: It rained for while every day we were there. The good thing about that is that it sent people back to their resorts. It wasn't cold, there usually wasn't lightening so it only really bothered us on one day. Use rain times to your advantage and stay in the parks.
Wet Clothes: Plan on doing laundry. I wasn't planning on doing laundry but nothing would ever get dry in the room and we got wet a lot! I ended up doing laundry so that we wouldn't have so much damp stuff hanging around. Make sure you have extra shoes for the same reason.
Lines were not too long early in the morning. We had a wonderful magical trip and I would do it again in a heartbeat!
Now, with all of that said, if I could plan any time in June I would probably go the 2nd week of June, about a week after school gets out here.
For a real summer treat though, you need to visit
Disneyland. We wore JACKETS at night and in the morning at the end of June and early July. We thought that it was hilarious that people around us were complaining about the heat in the afternoons when it hit 87 degrees!

We had a great time and the weather was unbelievable!