A common response to the moral argument is something along the lines of "I don't need God to tell me what's right". And, depending on the way this is meant, it's true. God has already written His law on our hearts, and so while that's still technically God teaching you the difference between right and wrong, it is only in the same sense as God having taught you to breathe.
But one thing most atheists forget is that, whether they like it or not, they are living in a society that has been heavily influenced by Christianity. There was a time in our past when God was a large part of our national identity. There are churches dotted all over the place, and it wasn't too long ago you would get funny looks if you didn't attend at least one of them.
By contrast, looking further back through history, we find that the values atheists most love - mercy, equality, the value of life - just don't exist like they do here in the post-Christian West. Pick any culture you like, you will see horrendous behavior practiced throughout. Pre-Christian history was not a great time to be alive.
Sadly, as time goes on, we are seeing a great decline in Christianity. The result? Our culture is seeing a great decline in moral standards, too. Things atheists used to claim credit for, even boasting that they must be better people than us because "you need a God to tell you not to do this horrible thing", they now commit openly, and celebrate publicly.
Thus, the response to the moral argument is no longer "I don't need God to tell me how to live a moral life", but "I don't need God to tell me how to act like myself, sins and all". Of course you don't need God to tell you not all humans are created equal, so it's ok to kill them in the womb, or just assume they're dangerous because their skin tone is lighter than others. Of course you don't need God to tell you the primary function of sex is pleasure, so you can do it with whoever you want, with consent being the only (loosely enforced) rule. Of course you don't need God to tell you every form of work is slavery, and it's the job of the government to provide for your every desire. You do all these things by your own sinful nature!
But you do need God to correct these errors. As previously noted, God has given us all a conscience, but He doesn't force us to obey it. And lacking that force, we all go astray, doing terrible things, not even realising how terrible they really are. Far too often, the only time we see sin for what it is is when it comes back to bite us. Even this does not always cause us to repent. We may think "I'm a generally good person", or "at least I'm not as bad as some people".
But ultimately, there is a judgement coming, far greater than any Earthly consequence we have ever experienced. It is appointed once for every man to die, and then face this judgement. At that time, every sin we just gloss over as if it's nothing will be brought into account. Thus, the need we have for God goes far beyond Him telling us how to live a good life. Ultimately, we need Him to account for every time we failed to do so.
Ultimately, there are two ways the final judgement can go. On the one hand, you can be judged entirely yourself. You can go in with all your good deeds, and all your bad. "But we are all like an unclean thing, And all our righteousnesses are like filthy rags; We all fade as a leaf, And our iniquities, like the wind, Have taken us away." (Isaiah 64:6). It's much like an Earthly trial, really. When you commit a crime, the only questions in the resulting trial are did you do it, and what should be done about it if you did? No one gets let off for a bad deed just because they did good. At least, not in a just system. How, then, can the ultimately just God overlook our sin?
By taking it upon Himself! And so He did. 2,000 years ago, Jesus, the Son of God became incarnate in human flesh. He lived a perfect human life, never once committing any sin, yet He died as a sinner should. On the third day after this, He rose again. This one act of love gives us, as sinners, a beautiful alternative. Rather than receiving due penalty for our sin, we can repent of it, confessing Jesus as Lord, and believing in our hearts God raised Him from the dead. With this faith, we will instead be rewarded for our good deeds, as if we had lived the life Jesus lived.