Yeah to live or work anywhere in the world that is not somewhere you were born or that does not have a very serious agreement (the EU has agreements that any country within it can have its citizens live and work in any other EU country, though it can get complex and you may still have to register in the country you are in even if it is just a simple formality, as a US citizen then you can do kind of similar things in Canada
http://www.canadavisa.com/canada-immigration-usa-citizens.html ) you are going to need various tickets. How annoying these are to get varies with country. In most cases it helps if you have a parent, spouse (the EU is not as big on marriage as US immigration is/was but it does not hurt), possibly kids and in some cases more distant relatives (some places will accept grandparents, or have a path for that). If you have some desired skills then that can work, and indeed is probably a bit easier in many cases than the US equivalent, and if you have an actual job offer then great. Afraid you are on your own to find what goes there and it varies by country and even region within it, though languages, medicine (doctor can be easier but nursing shortages are far from unheard of), teaching, some of the less common sciences, some kinds of mining are good bets.
If you are going to retain US citizenship you do still have things you are supposed to do as well -- I believe you are supposed to continue to file some form of tax return, even if you have not set foot on US soil the entire year.
http://www.visalogic.net/uk/work-permits/4/124 looks like a good overview for a lot of the UK stuff. If you are not going to become a citizen of a EU country then US citizen working in ?? is a good search, here is one for Germany
http://germany.usembassy.gov/acs/working_in_germany/
Also while I am saying EU it does not necessarily mean what you think of as Europe (for instance Switzerland is not in the EU)
Also to be a pedant I do have to point out in the US a green card is not citizenship and will allow the owner of it to do basically everything that is not serious government work (you won't be able to join the FBI sort of thing). It does need to be renewed but unlike extending a work visa it is usually a matter of paying for someone to rubber stamp a form and also unlike several flavours of work visa you can change jobs as readily as anybody else.