I flee the U.S. to have all my medical related tests and work, out of pocket, in the E.U. And it is cheaper and much much better experience. (And I don't get different treatment compared to any members of my family that are insured in Europe for paying out of pocket.)
First of all: I speak with a doctor. Not a nurse, an administrator to size me up, to see if I am in actual need of an appointment, but a doctor. (Yes, this has happened to me in the U.S. I find it unacceptable, especially given that I was apparently in much more dire situation than I even thought, and was lucky to be seen by a doctor, otherwise I would have joined the disabled group of individuals.)
I am not sure why people in the U.S. keep bringing up the UK [Edit: -- not sure if that is what you are implying but most people are in other comments]. Pick any EU country. Sure, you might not have a 5 star doctor's office, but you are going to be treated by a doctor efficiently. And that is what matters. Don't waste money on administrative tasks and fees.