I can tell you that you cannot become a certified planner with local government in Great Britain with an American planning degree. Their standards require a Masters Degree from a selected number of UK universities. That is not to say you cannot work for a consultant, however (be warned)the legal and practical aspects of planning are substantially different in the UK.
Since a planning degree from an American university is based on American laws/regulations/constitution, etc., it is not so easy to just get a planning job in Europe. Design (landscape architecture or architecture) might be more transferrable.
In addition, if you are not a citizen of a European Union country, it is very difficult, if not impossible, to work in the EU.
I would suggest going to a planning school in the country you would like to work in. Yes, it will be more expensive than public universities in the US, but you will end up with an education that will be applicable in that country, and you will make professional contacts that will most likely lead to jobs. Of course, there is still the question of citizenship, which may still pose problems in getting the job you want.
Another option would be to work for a big American consulting firm that has offices and/or projects overseas. EDAW is one that has European offices, but I hear it is pretty difficult to get a foot in their door. There is also Parsons-Brinkerhoff. There might be more.
Or, you could go the NGO route, but these are mostly in developing or Eastern European countries, not in western Europe.