The Euthyphro Dilemma is basically asking the question "Is an action Good because God said so, or does God say so because it is good?" Or from the other direction, if an action is immoral, is that because God said it is immoral, or is it inherently immoral?
As an example, let's think of something that we can all agree is horribly immoral, say child rape. Assuming we all agree this is immoral, there are 2 options here
1. It is immoral because God said so
2. It is inherently immoral independent of God
If it is option 1, that it is immoral because God said so, then what would happen if God changed his mind? Could this suddenly become moral? Could God command you to go rape a child? Would this make it a moral action? If the only reason that it is immoral is because God said so, then if he changes his mind then it must be moral.
The common response to this in my experience is to say that God wouldn't do that. This simply avoids the question and in my mind, puts you squarely in answer 2. Why would God never do that? It sounds to me like the claim is that he is moral and that is an immoral action. Therefore God would not have you do it. But if morality is defined by what God tells us to do this makes no sense.
What about option 2? What if morality is just what it is. Child rape is wrong. Period. If God told you to rape a child it still would not be a moral action. Perhaps you would say that since God is moral he would never do that, but the reason for that is what I am interested in. Since he is moral and child rape is inherently immoral, God would never tell you to do that. If this is the case, there is some morality outside of God.
If there is a morality that is independent of God, the charge that atheists are inherently immoral falls flat. We don't need God to tell us how to be moral, because morality exists outside of him.