This is actually the most important thing. Have an awful vocabulary, a rudimentary grammar, but get the pronunciation right. You can always buff up on your words or rules, but getting rid of an atrocious accent is very hard to do. Utterly impossible if you're Styrian.

I wonder whether singing helps. The overall musical structure might make proper phonetics easier to memorize; on the other hand I've heard quite a lot of people who can sing without an accent, but sound very differently when actually speaking. Maybe it is too much rote memorization.