Some states have automatic appeals for those who have been convicted of 1st degree murder and sentenced to death, other states I believe leave it up to the defendant to decide.
A question I have for RJers is, if there is absolutely no doubt that the defendant committed the crime, do you think there should even be an appeal? I'm thinking of a few high profile cases in recent years where the guilt of the defendant was proven conclusively, yet they may sit for years on death row.