I think it differs a great deal between countries in the EU. I'm Swedish and I have an Amex card and most of my friends and colleagues use Amex as well. I use it for about 85% of my spending each month. The only places in Stockholm that don't accept it are smaller coffee shops, restaurants that don't get that many tourists and mom and pop stores.

You could probably get by here with just an Amex card and cash, but I keep a backup card for the places that don't accept Amex.