We almost always go the first or second week of June. It's hot, but what's worse that time of year is the humidity. Expect to have to peel your clothes off of yourself. It does rain, frequently, but I've never felt the need to carry ponchos. You're already soaked in sweat, so honestly, being soaked in the rain is a step up from that. Plus the rain kills the crowds, so as long as there's no lightning to shut down rides, the lines become 10x more manageable.
But, if you do go in the summer, June is still MUCH better than July/August. In my personal experience, it's less crowded (that's not to say it's not crowded, but the crowds are usually better than mid- to late summer), and while it's hot and humid, it's not deathly so, like it is closer to August. Then again, I grew up in the south, in a state that can get almost as nasty as Florida in the summer, so if you're from a cooler climate, it may all feel miserable to you