Richard Dawkins and Christopher Hitchens discovered as their book tours brought them through the very buckle of the Bible Belt that wherever you go, there are plenty of people who are happy to be and know atheists. But you're right, finding them can be a pain if they don't come to you.
Honestly, it's worth bringing up your atheism early if you're worried it will end a relationship. Sure, it might last longer if you don't, but you'll always be worried.
My first port of call would be the internet. www.richarddawkins.net has a social section, there's regularly an ad on the right of this site for www.freethinkermatch.com, and many of the conventional dating sites have an option to search by religion. (Yes, I know atheism isn't a religion, but having it on the list is better than nothing.) There are lots of other communities out there.
Offline, just go out and meet lots of people in any setting which isn't explicitly religious. Don't expect everybody to be devout, because trust me, they aren't.
Finally, don't think you can only find happiness with another atheist. My girlfriend is a liberal Christian, and although we disagree on many things we're happy in an atmosphere of tolerance, personal respect (as opposed to respecting all each other's views) and free exchange of ideas.
- SmartLX
Do a little research on the Golden Rule to see just how exclusively Biblical it is. Not.
The Bible has plenty of instructions that everyone would agree are great ideas, like "thou shalt not kill". It also has plenty of instructions everyone would agree are terrible ideas, like much of Leviticus. The good ideas are not only common sense, but they can generally be found in other sources pre-dating the Bible by centuries.
Consider the possibility that the good ethical advice in the Bible is not good ethical advice because it's in the Bible, but rather its human authors put it into the Bible because it's good ethical advice. It's a reflection of human nature, much of which is benevolent.
This may make the Bible a good moral guide if you cherry-pick the sensible parts, but it certainly doesn't establish its God as the only source of morals.
- SmartLX