I've been in June before and it is crowded, but you can definitely have fun. As the others said, you will only feel "concert crowded" after fireworks and parades.
Epcot is probably the best park for crowd control. It's really big and open so it never feels packed. AK is the worst. The paths are kind of narrow and you have to backtrack a lot to get around, so the paths always feel crowded, even in the slow season.
Go with a plan and be okay with the fact that you won't ride everything (though, it's hard to ride everything even in the slow seasons). The summertime has the highest percentage of "clueless" travelers- people who show up at 11am and wander the parks looking at their maps saying "What should be do next?" A knowledgable traveler should be fine.
I'd check out Tour Guide Mike. I have yet to use him, but I will once I start going in the summer again.