Troy Davis
Troy Anthony Davis was a 19-year prisoner on Georgia's death row who was sentenced to be executed on September 23rd, 2008 for the murder of a police officer. The Supreme Court suspended the execution.
The problem was that there was no conclusive evidence that he actually did it. Seven out of nine of the state's witnesses have either recanted or contradicted their testimony against him, frequently citing police coercion, and no murder weapon has been found. One of the two who didn't recant was later implicated as the actual shooter. Unfortunately, despite all of this, he was still on death row.
Exposition
There is also an important general principle involved with state executions. We can never be absolutely certain that we are executing a guilty man. From start to finish, government controls the whole process, from police work to trial to sentencing.
For an innocent man to be caught up in the process is bad enough, but having the state carry out an execution as well is intolerable. No amount of safeguards or appeals can prevent government corruption or colossal errors, and if the wrong man is executed, there can be no redress of grievances. The death isn't the only thing that matters. Prisoners suffer terribly on death row.
Update
Troy Davis was executed on September 21, 2011. His time of death was 11:08 PM EDT.