As an American who tries to travel alone as much as possible (I simply prefer it) I say you'll have no problems at all.
First off, the world isn't the American-hating sea of bitterness that the news makes it out to be. People tend to hate our government, not us. If anything, it's our own people trying to make it harder to travel and be abroad. Other countries welcome us (and our dollars) quite readily.
Second, the American travelers that do get mocked and talked down to are the typical tourists/package vacationers/loudmouths, not backpackers who are perceived to be relatively conscious and easy to get along with.
Third, most foreign folks are cogniscant enough to know that the Americans they do meet abroad are part of the shockingly low 18% who actually DO travel overseas and aren't the gun-wielding psychos they see on TV.
Sure, you might be judged right away when you tell people you're an American. I have, and it hasn't bothered me. Within 10 minutes or 2 beers (whichever comes first), everyone I have met with pre-conceived notions about Americans has either erased those notions or claimed me as an exception.
In fact, you will see some "Ugly Americans" here and there (especially in Amsterdam, these kids act like they've never seen weed or half-naked girls before) and you may well get a ribbing here and there, but just take it as it comes and be yourself so you're judged on who you are, not where your passport says you're from.
Of course, I say this ideally and all, when in fact, I try to avoid fellow Yanks while I'm abroad