Same here as a french guy. Even worse is that I can usually hide it pretty well (or at least diminish it) until I have to pronounce the letter 'R'. Damn R's...
What would be the equivalent french show for going the other way? I found Braquo on Hulu and started watching/translating/studying it, but realized pretty quickly that most of what they say should not be said in good company.
Try "Un gar une fille", it deals with typical, every day situations so it's good for vocabulary. Unfortunately, it may not be suitable for beginners. The conversations are fast paced and I haven't been able to find any French subtitles for it.
I think it is dependent on which accent you have. In my experience as an american (who has met a lot of foreigners), I have found that germans and the dutch seem to have the least problem with the american accent. (Australians too, for a non-european example). There can be a number of reasons for this to be the case, but this is what I have noticed.
