I quickly found there are at least three ways visiting U.S. pilots can access general aviation aircraft in Europe. The main ways are through aero clubs and conventional rental arrangements. The latter includes both pilot training organizations and organizations analgous to small fixed base operators (FBOs) in the U.S.
If you have access to a U.S.-registered aircraft-one sporting an N-number-you have another way to fly in Europe. Technically, if the N-registered aircraft is not permanently based in Europe and you are not a European Union citizen, you can operate it as pilot-in-command with your FAA pilot certificate (although I wouldnt advise this unless you do some careful preparation and planning and fly initially with a local instructor). See this articles main text for why.
The airports I flew from included Buttwil, Switzerland (LSZU); Trier, Germany (EDRT); Luxembourg City, Luxembourg (ELLX); and Chateau-Thierry, France (LFFH). Luxembourg was the only airport with an operating tower. Both Buttwil and Chateau-Thierry had grass runways only.
I flew Cessna 172s, a Cirrus SR20, a Piper L-4 (military version of the J-3 Cub) and a Robin DR400. The cost of using these airplanes ranged from the equivalent of $177/hour (L-4) to $355/hour (Cirrus).