Operating N-reg planes is not big issue, the only thing you need is the respective permission to enter each country (usually issued for 6 month, the old 1 year permits are qute rare nowadays) - but, as there is no country authority "Europe" you have to do this for each separate. So, travelling across Europe with N-reg can be tricky.
Obviously N-registered aircraft with a standard ICAO compliant FAA airworthiness certificate require no such thing.