I opened my bank account in my married name using my marriage certificate...the name on my passport wasn't (and hasn't) been amended.
If you want to get it changed, you can send your passport to the US embassy in London. The service is now free, though you have to pay postage, of course. I think most people get theirs back in a couple of weeks. They just make a note in the passport that your name has changed -- nothing happens to the page with your name and photo.