Maybe some theists say that you can't be moral without God but I think the actual argument is by what authority can one declare morality or immorality other than God?
Any discussion of morality always stumbles on the question of authority.
There is a paradox in human nature that demands, "Who said so?" and the response is always "Because I said so." It's easier (and more convincing) to cite some great authority/rule giver/Godhead/PYL, than to argue that some action is right or wrong.
The problem of accepting a moral code as dictated by an independent authority is the code has to be codified. This seems obvious, but it means the code is frozen in time. There was a time when stealing a sheep could mean serious hardship for the shepherd and his family. The fact it was difficult to prevent theft compounded the problem. Sheep thieves were put to death, mostly for the deterrent, not because sheep were that valuable. It was a reasonable solution to the problem.
In a modern society, we have a huge police system to deal with thieves and for the most part take the burden of stopping thievery off our shoulders. As a result, killing a thief seems overkill. If we stuck with the written code, car thieves would be put to death.
Our comfort with citing authority for our moral decisions will always push us to do things which no longer serve the purpose of a moral code, which is to provide a set of guidelines for people to live in peace, safety, and cooperation with one another.