The DCT was challenged by Plato in his dialogue, Euthyphro. In this dialogue, Socrates asks essentially this question:
This may provoke a reply to the effect that God would never command such things, because God would never command what was wrong. However, this argument cannot be made if the DCT is to be maintained - under the DCT, if God commanded something, it would not be wrong.
Secondly, if God commands an act because it is good, this again undermines the DCT, as it means that the act was good independently of God's commanding it, and therefore being commanded by God is not the only reason the act is good. Rather, whatever reason God had for commanding it is the ultimate reason that it is good.
This line of attack on the DCT is well-enough known that it is referred to as the Euthyphro dilemma. Plato is generally believed to have refuted the DCT outright. However, it should be noted that certain other theories that link morality to God are more subtle and are not straightforwardly refuted in this manner.
Another problem for the DCT is what to do when there is no command that is relevant to a particular ethical dilemma.
For example, the following is an extract from a Christian website's review of the movie Alive, and its portrayal of cannibalism:
Plato's Euthyphro
The question is such that either answer seems to lead to the rejection of the DCT. Firstly, if an act is good solely because God commands it, then that would mean that if murder, rape or theft were divinely commanded, they would be good. This seems to be absurd, although on some occasions it has indeed been seriously proposed.Missing commands
This illustrates the problem - a follower of the DCT wishes to do God's will, but if he only has access to specific commands, rather than general guiding principles, he will struggle when faced with ethical problems not covered by God's commands. By contrast, other ethical systems (especially utilitarianism and Kantianian deontology), lay down general principles for ethical action which (at least in theory) allow a person to deduce the right course of action for any situation.