I'm asked this question a lot whenever discussing the moral argument for God's existence. Usually the argument goes something like this:
1. Every law has a lawgiver. (Premise)
2. There is an objective moral law. (Premise)
3. Therefore, there is an objective moral lawgiver. (From 1 and 2)
The moral lawgiver is then associated with God. Now, whether this argument is sound or not isn't the point. Rather, what I'm interested in is answering the question contained in this post's title: would I suddenly become immoral, e.g. raping, murdering, etc., if I abandoned my belief in God?
The answer is obviously no. The follow up retort is usually: "then objective morality doesn't depend on God."
This objection is fallacious for the simple reason that it confuses moral epistemology with moral ontology. What the skeptic should say is that "objective morality doesn't depend on belief in God." This is quite distinct from the former assertion. The reason I would remain moral (assuming I'm considered a moral person) even if I abandoned my belief in God is because I would still be created in the image of God. It's just that my hypothetical atheism would be inconsistent with my recognition of an objective moral law.
Readers are free to disagree with the moral argument all they want. In fact, I encourage open debate. However, it's important to understand what the moral argument claims and what it doesn't claim.