By "Buddhist" countries I do not mean just countries with a large number of Buddhists, but countries whose governments operate according to the principles of Buddhism. These countries do not have the death penalty.
Why?
The most obvious reason is the First Precept which says, "do not kill."
The concept of revenge is totally foreign to Buddhist mindset. Buddhists believe in the Law of Karma, not in the law of an eye for an eye.
But there is a more subtle reason not to execute violent criminals. According to Buddhism, the conditions of the next life time (reincarnation) depend on the last thoughts, the last state of mind of the one dying. A violent criminal's mind is not exactly in a peaceful state. Subjecting him/her to an execution is going to make his state of mind even more violent, fill him with anger. As a result, he is going to be reborn even more violent than he is now. And no one is going to know the newborn baby is actually the guy who murdered the girl next door (or whomever). So, he is going to grow up, completely free to commit even worse crimes than he did before (has anyone noticed that crimes in countries with the death penalty are getting more and more violent as time passes by, has anyone wondered why).
Hence, it makes more sense, from the Buddhist viewpoint, not to execute criminals, as doing so only increases the problem. It makes more sense to keep them alive and give them a chance to come to a realization that what they have done is not a good thing. That gives them a chance to mature and grow, and eventually become non-violent. Even if it takes them several life times, it does speed up the process of growth and understanding, while killing them slows the process down, and makes them worse than before.
That is the main reason Buddhist countries do not have the death penalty.