I'm a consistant first or second week June traveler. I'd pretty much echo deirdrefitch's opinions. The rain comes, but it's quite frankly a relief when it hits. Unlike some September storms I've seen that send you underground for a couple hours, the storms in early June seem to come and go in a half hour. It's actually kind of funky. I've seen pouring rain through sunshine, quite the "Magic" site.
A warning: Saturday's in MK, never a nice thing, can be pergatory during this period, specifically on Saturday June 7, which is Gay Day in the Kingdom. A day to be avoided, crowdwise. Beyond that, we avoid early entry, use fast pass, hit the E-Tickets early in the morning, and find most of our waits are fifteen minutes or less. A suggestion: get there early, take a break in the afternoon, come back when a good portion of the crowd is either leaving or eating.
Pat