Suppose for a second that the US was at war with Canada and John Doe was an American citizen, but he was serving as a uniformed combatant in the Canadian army. Under such circumstances, I can't imagine anyone would have a problem with the United States military killing John Doe in battle without trial.
Obviously, the circumstances around the kill list are murkier... The "war on terror" isn't a war in the traditional sense, terrorists or suspected terrorists are not uniformed combatants, the CIA is not necessarily the military, and it's not really certain that dropping a bomb on somebody's "cafe experience" is equivalent to killing them in a pitched battle in a declared war.
Still, I have a hard time imagining what rights a US citizen is supposed to be entitled to when they are effectively a foreign terrorist working to attack the US. Are we supposed to have a trial for them without them present? (We all know how that would go.) Ask for their home country to extradite them? (Country will almost certainly either be unwilling or incapable.) Send the Navy Seals to kidnap them and force them to stand trial? (Dangerous and wildly impractical, not to mention boots on the ground increasing the risk of creating a larger conflict with a foreign nation.)
Personally, I have two take-aways on the issue:
1) I wouldn't want to be the judge that has to sort this stuff out, and I hope they do it well. It's not easy.
2) I think the bigger problem is drone killings in other nations FULL STOP. Regardless of who the targets are, it's completely unacceptable for any country to think they can just fly into another nation's airspace and start dropping bombs on anybody they please without a declaration of war, some sort of cooperative MOU, or a join military exercise...