My first thought is "Who cares?" Does it really matter where morality comes from? Say it comes from a deity of some kind. That still does not mend any of the other myriad problems that I have with theism in general and Christianity in particular. But I don't mean to avoid the issue.
So my second thought is that theists are wrong when they say that without a god there is no explanation of morality. There is an explanation, but they simply do not accept it. My explanation is that morality is not objective and does not point to any kind of universal truth but is an adaptation and a byproduct of civilized life. Just as "Logic" is the language we use to make sense of the physical universe, "Morality" is the language we use to make sense of social interactions. It is far from the theist trope that atheists must all be seething with a desire to rape, murder and steal.
"Why don't you just kill whomever you please?" theists ask. "If there's no moral standard, then what's keeping you from shooting me in the head and taking my money?" I don't do these things because I don't want to do them. And why not? Well, firstly I don't have an intense desire to kill or take anyone or anything. We humans are sociable and while we can easily point to deviants (serial killers, etc) I would say that those exceptions prove the rule rather than otherwise. If humans were all bloodthirsty monsters, we wouldn't notice someone like Charlie Manson. And beyond the actual desire to kill, there are rigorous social institutions in place to keep these things from happening. It is unhealthy for a civilized society to allow murder and theft, so it finds ways to keep those under control. Even if I wanted to kill someone (and I don't, I assure you), I would think very seriously about it because I would very likely be caught and punished severely. It is very likely not worth the trouble.
Rape and murder are extreme examples. Let's discuss theft and dishonesty instead. In general, telling the truth is a good policy. This keeps social interactions functioning efficiently. What would I do if I found a $20 bill on the side of the road? I'd look for someone to whom it might belong. If I found no one, I'd keep it. What if I found myself alone in a parking lot staring at a convertible with the top down and a wad of bills rolled up in the cup holder? No one could see me take the money, no one would find out, but the owner of the vehicle would come back to find $100,000 gone. I wouldn't take it. It isn't mine, and more than that, it is someone else's. I would hope that someone else would respect my property the same way. This is the trust that we need to have in order to keep our world from tumbling into anarchy and Golding-esque savagery.
It makes more sense to me that morality is something that we all work for, a standard that we all hold up together. This way we can constantly raise the level of our social ethics. If morality is an exclusive dictate from god, then it is also arbitrary, contingent on his pleasure. If god dictated that toes were an abomination, then suddenly toes are immoral. Theists insist that it is far from arbitrary precisely because it is from god. This is why the argument for the existence of god from morality doesn't make a convincing argument.
I am content to have my morality be flexible and plastic, dependent on the situation. I am happy that our social morality has evolved beyond our primitive roots. I am glad that it is continuing to evolve, that the bar will continue to be raised. I hope that one day, centuries from now, future humans will look back on our time and remark how vicious and evil we all were, just as we look back on good people from throughout history - those who cheered at gladiator arenas, those who owned slaves, those who beat women. One day we'll respect the rights of all animals, caring for the earth and all of those who reside on it with equal care. We'll never get there if we're always striving to reach a moral code that is centuries behind the times.