I guess I never think about it. I'm state side for a bit now, but when living in Edinburgh, I never had any problems with my accent, or with people pointing it out to me all the time. Of course, Edinburgh is a pretty cosmopolitan city, so there are people from all over the world, living and working there. I was most certainly not the "token" American. If you are in a small villiage or somewhere that few American's live (or visit) people may get a kick out of it, but as long as you are not loud/brash/obnoxious, you will be just fine.
I know you are getting married, so it may not matter to you, but many people (including guys) will think your "exotic" accent is quite sexy.