The animosity against the US comes directly from our intervention into the economic matters of other people. In supporting our economic interests we have successfully managed to piss of quite a few people while hiding under the rhetoric of peace and freedom. Remove that intervention and the overwhelming majority of the animosity disappears. Radical Muslims aren't the only ones with an axe to grind against the US. As somebody who has traveled to a few counties, I've realized in a lot of the countries that have a bad opinion of the US, we've intervened in favor of their oppressors. Cuba, Liberia, El Salvador, Guatemala, Ethiopia, these are just some of the countries where people really do not like the US and one can't blame radical Islam. The only difference is the fervor and the ability to express their rhetoric. Whereas the Middle East is simply flowing with a pretty healthy base of people willing to fund terrorism, South America simply does not have the social infrastructure to do this. The elites in these nations put wealth before religion so the attacks are directed towards them. Bogota, Colombia was known as the car bomb capital of the world only a few decades ago. The rich in these countries are constantly being kidnapped for their support of American economic policies.
In the end, radicalism in the Middle East boils down to US intervention in the political and economic matters of the region. Our support of the Sha of Iran, Saddam Hussein, Mubarak and various others dictators has made Middle Eastern youth fall back on religion because they are totally disenfranchised. Sadly, there is a political and economic elite which has been disenfranchised right along with them and are more than willing to fund terrorism.