Been there dunnit. If you want to fly around cheap then go get yourself an FAA ticket, it mey also come in handy as a way of flying in Europe on N reg aircraft, especially with a US IR. multi time much cheaper over there, and if you become an instructor expect loads of hours and plenty of flying other people's aircraft at their expense when actual IFR grounds the Doctor's Bonanza.
All great fun, and if you can get actual European IFR experience as well so much the better.
Problems? There are several potential ones, but it depends on your personal situation and mindset whether you let yourself fall into the various pitfalls. Most of them have to do with the way that procedural flying is taught in the US, indeed lets extend that to the way that flying is taught in the US period. There are loads of ways to fly aeroplanes lets face it, but the requirements for the Cpl and IR over here are stringent in a different way to the US system. There are advantages in both systems, but I know several extremists on both sides of the Atlantic who swear blind that the people on the opposite side are accidents waiting to happen.
If you can accept a flexible approach then you'll have a lot of fun.