Jojo the Swede writes: Swedes have had great success as directors (Lass Hallstrom) and especially as actors (Ingrid Bergman, Alexander Skarsgard) in the US, but how is it with writers? I have long dreamt to write for a TV drama series. The thing is, I'm Swedish, so the alternatives are the PBS of Sweden and BBC, but really - everything good comes from your side of the Atlantic. People here know more about America than their own country! Is there any interest in Europeans (who are not native speakers, as the British and Irish)? Do we have a harder time proving ourselves worthy?
I think the main point is that it's all about your script. Write a great script. I don't think your background will hurt you at all, though the fact that you're not a native English speaker may mean that you'll have to work harder to write eloquent scripts in English. Often your unique background or specific experiences can help you, and producers/executives/etc. definitely like to find fresh, original voices.
If you want to work in the American TV or film industry, your other main obstacle would be making contacts and getting noticed here. It's not easy to get an entry-level job to make connections since it's so tough to get a company to sponsor your visa (I've blogged about this before). You might try to pursue some kind of writing career in Sweden first.