Those on all sides of the religion debate are familiar with Dostoyevsky's line, "If God does not exist, then everything is permitted." This is another expression of the argument from morality; i.e., the existence of morality implies the existence of God. "Morality implies God" and "No God implies no morality" are logically equivalent statements. If you don't understand why this is so, type the word "contrapositive" into your search engine of choice.
Many people have attacked the argument from morality with various logical devices, but that's not what I want to get into here. I'm more interested in approaching an accurate understanding of reality than in scoring imaginary debate points. Let's look at the parts of this argument. Is there such a thing as absolute, unchanging morality? I'm not sure there is. I think what people are really saying with statements like Dostoyevsky's is, "Without religion and its underlying fear of punishment in a life beyond this one, society would descend into chaos."
I think most of what we call morality entails rules of the road that have been deemed useful to society at some point. Since humans are social animals, we prefer to live in groups. This means that something like random killing is a survival threat to the group, and other members of the group are therefore motivated to stop it. So there are perfectly natural explanations for why we have rules of order in society.
Even those who claim morality is fixed and can be found in, say, the Bible are on shaky ground. If this were true, then people's understanding of the Bible wouldn't change over time. But it obviously has. In the past, the Bible has been used to justify slavery and poor treatment of women and gays, just to name a few. Now you don't hear pro-slavery preachers, and things have at least gotten better regarding treatment of women and gays in many religious groups. Is this because there's been a new revelation from the heavens where God says, "Maybe you shouldn't use me as an excuse to be such a dick to people?" To my knowledge, they're not selling Bibles with new bonus books added or anything, so I'm going to say no. What's changed over time, then? Human understanding, that's what.
This doesn't mean I can say, "No morality implies no God." The rules of formal logic don't work that way. But I can say that morality can't be used as an argument for God's existence because it hasn't really been established. You can't use an unproven idea to prove something else. Perhaps it would be nice to live in a world of absolute certainty, but there's no evidence that that's how the world operates.
Some Hindu traditions emphasize the phrase "Neti, neti" regarding the Divine. This can be translated as "neither this nor that" or simply "not this, not this." In other words, if you think you understand what the Divine is, you're probably wrong. This concept can also be useful in observing earthly matters. Anytime you see certainty, it probably wouldn't hurt to step back and say, "Well, maybe not."