As far as most of Europe goes I don't think they "hate" the U.S. I think it is more that Europeans are building a common society not as Europeans, but as non-Americans, much in the way that John Kerry ran for president in 2004 - not on any solid Democratic plan, but as a non-George Bush. It's convenient to point fingers at America and blame her for all the problems in the world. Americans are blamed for a myriad of issues, from global warming to cultural decay. As Jean Francois-Revel (a French politician, journalist, author, prolific philosopher and member of the Académie française since June 1998) puts it:
The fundamental role of anti-Americanism in Europe in general, and particularly among those on the Left, is to absolve themselves of their own moral failings and intellectual errors by heaping them onto the monster scapegoat, the United States of America. For stupidity and bloodshed to vanish from Europe, the U.S. must be identified as the singular threat to democracy (contrary to every lesson of actual history). Thus, during the Cold War, it was dogma among Europeans from Sweden to Sicily, from Athens to Paris, that the "imperialistic" power was America, even though it was the USSR that annexed Eastern Europe, made satellites out of several African countries, and invaded Afghanistan, even though it was the People's Republic of China that marched into Tibet, attacked South Korea, and subjugated three Indochinese countries. A similar dynamic applies today in the war on terror.
Twice in the twentieth century, Europe nearly committed suicide. Both times Americans jumped in to bail them out, at huge personal and national cost, something which must eat at their psyche - the colonial upstarts who dared to create a new world had to come to the aid of the old. Those classless, cultureless, money-grubbing Americans had to go in and do a managed intervention, keeping Europeans safe from themselves and a now-collapsed Soviet threat. (That American money, while considered to be dirty capitalist spoils, also comes in handy, too.) America's steadfast support of Israel, is another bone of contention.
Now, that is not to say EVERYONE in Europe hates the U.S., and to say, or even imply, that is the case would be very unfair. But to say or imply that "it is all in our heads and that there isn't a level of, dare I say, hatred, for the U.S. is equally unfair. Europe aside, there are plenty of places with extreme hatreds for the U.S., the Middle East being prime among them.
"The tree of liberty must be refreshed from time to time with the blood of patriots and tyrants." -Thomas Jefferson
"A man-of-war is the best ambassador" -Oliver Cromwell