It is important to note though that there is no such thing as an EU Visa (or Work Permit for that matter). While, as Vicky pointed out, the Schengen region as a whole all have visa free visitor travel for US passport holders, it is still up to the individual countries to decide who they allow in and in what fashion.
Also, realize being a resident, even a permanent one of a country does not normally allow you to travel as if you have a passport from that country. So as long as you don't have a UK passport, you will be treated in other countries as if you are from the States (assuming you are a US passport holder). My passport has the stamps to prove it!