Does anyone know of any film festivals with a May or June Submission deadline?

http://www.withoutabox.com is the ultimate submissions resource these days - pretty much every fest is using it for expediting the submission process, and it's free for you (of course you still have to pay the submission fees to the fest).

They have all fests sorted by upcoming deadlines.

Try the Festival Directory at IndieFilmmaker:

http://www.indiefilmmaker.org/film-festivals.asp

Thank Imran. That's a great site!