'Without God,' goes conventional wisdom, 'morality is just convention.' And, to expand on that conventional so-called wisdom: in a godless universe with godless believers, anything goes; morality is relative, a matter of conventions relative to a particular society.
Now, I know little of psychological researches, but philosophical reasoning and the need for consistency can debunk that so-called wisdom.
First, the claim that morality is relative and conventional is itself a moral claim - yet it is put forward as an objective and non-conventional fact. Minimally, this shows that not all morality is relative, whether or not there is a god.
Secondly, people who put forward such relativism often draw the conclusion that therefore we 'ought' not to interfere in the practices of other cultures. But if morality is all relative, then there can be no objective 'ought' about not interfering in other cultures. Relativism does not offer support for toleration. If all morality is relative, then toleration is no more objectively the answer than authoritarianism.
Thirdly, do religious believers really think that if there were no God or gods, it would be perfectly morally acceptable - objectively - for them to rape, pillage and murder? I doubt it.