I think even once you eschew God and the bible, unless someone challenges you, you'll still rely on your basic moral code which you got when you were younger (and probably indoctrinated). A lot of that might evolve into what "feels" like the right thing to do, more than what the bible said.
Then there are those who think about this sort of thing all the time. I'm not one of those people, normally. I don't like philosophy very much. I prefer science and testable claims to pontificating about things that can't really ever be known. But as I venture on my Quest for Knowledge, lately I've run into philosophy more often.
I've even subjected you to it, such as the whole Free Will vs Determinism argument (which I still have plans to continue).
I think we are best served with examining our ethical code and reevaluating what's in there and why. I did that some time ago with my personal Principles. It was a challenging but rewarding exercise.
Side Note: The Rationally Speaking podcast recently interviewed Joshua Knobe about his new field of Experimental Philosophy (x-phi), which was fascinating. Joshua even made them (and me) gasp with the results of one study that was very surprising! Here's a link to the episode.
The book club I'm in, as I've mentioned, is going to be reading The Moral Landscape by Sam Harris. I lost interest in our current book, so I've started in on this one and so far Sam Harris is speaking my language, that science can determine values for conscious beings.
The following excerpt from the book shows what moral relativism basically is (the woman in it is definitely a moral relativist), and why it's clearly flawed. This is an anecdotal story from Sam. But I think it's quite important to share:
Excerpt from The Moral Landscape: How Science Can Determine Human Values by Sam Harris. - Highlight Location 765-83 (chapter 1)
We already have good reason to believe that certain cultures are less suited to maximizing well-being than others. I cited the ruthless misogyny and religious bamboozlement of the Taliban as an example of a worldview that seems less than perfectly conducive to human flourishing.