If the answer is "yes", then you shouldn't have any particular problems, supposing that your skills (including English language skills) are competitive-- "any particular problems" meaning "no problems related to your EU citizenship, just the usual problems that people have trying to find jobs in a difficult market."
If the answer is "no", then you're shit out of luck. There's absolutely no way you're going to get an offer without a green card, unless your skills are absolutely stellar-- and your question refers to you as "your average coder."
On the bright side, there's a whole lot of countries where you can go and get work-- a lot more than there are for the average USA resident. I've heard the that there's quite the scene going on in Cambridge, for instance. And, at my last start-up, here in Norway, we hired from all over Europe.