We're already living in a world without God. Believers just don't realize it.
In any case, God doesn't make the rules. People do, and at least in a democracy, those rules are generally based on some concept of the common good.
As to why we'd obey those man-made rules, there are a number of reasons. Simple respect for others, for one. It's in the best interest of our society for another. Enlightened self-interest for yet another. And if none of those work for you, there's the fact that the government has the authority to put you in jail if you don't obey the rules.
Granted, the rules don't always make sense and aren't always fair (they're made by imperfect humans, after all), but in a democracy, there are ways to change or eliminate bad laws.
Contrary to what many Christians believe, the US was not founded on Christian principles. It is and always has been, in theory and intent if not always in practice, a secular nation. Our government is expressly forbidden by the First Amendment from either endorsing or suppressing any religious belief. Unfortunately, certain neoconservative elements in our society and our government seem bound and determined to change that.