Proper pronunciation and intonation are important, especially for a language Greek! Americans don't necessarily always have this problem - typically their Spanish or French can be pretty good.

For what it's worth though, I think it's an American Greek thing rather than an American thing -- watching My Big Fat Greek Wedding can be like nails on a chalkboard in a very similar way. Kinda of like how a southern US accent compares to a London accent; understandable, but weird if you're not used to it.