Dual citizenship

[Skunks]

My passport expires in May, and I definitely intend to renew it. But thoughts of renewal, plus a recent trip to France and a tentative (unlikely) trip to Crete have given me renewed thoughts of living for a year or two in Europe. As I understand it, the only way to live and work in a country is to A) live there already or B) have a long-term visa, which mostly requires you to be a student or have your job before you get there. Given that I really have no remarkably marketable skills, B is out.

However, my father is a first generation immigrant from Greece; according to the Greek Consulate website, paternal decent up through the grandfather is one route to Greek citizenship. I figure that Greece, being in the EU, would then give me access to most other European countries.

This is all very loosely researched, and all very vague & unclear to me. Are there any significant drawbacks to dual citizenship? I understand that living & working in a foreign country I would still be taxed for that money in the States. Is the reverse also likely to be true? I understand Greek citizenship requires some sort of mandatory military/social service (I also understand that most Greeks have some hookup that lets them avoid it); would that interfere with my American citizenship?