I think the problem is that children aren't being taught how to resolve their problems. Let's face it, running to teacher or to mommy when someone's bullying you either gets a response like "just ignore them," or else the parent takes over dealing with the problem instead of helping the child learn how to deal with it themselves. Add to this our popular media, in which the "bad guys" are most often defeated by being killed or beaten up... now, I'm not jumping on the "TV is evil" bandwagon. What I'm saying is that if a child isn't learning the conflict resolution skills s/he needs from a healthy source (parents, teachers, good role models, relatives, etc.), then s/he will learn them from whatever is available. And that most often means the media.
Then, there's another factor... some parents don't just neglect to teach their child non-violent means of solving problems. Some parents (by the examples they set as well as the things they say) teach their children that violence is
the answer. How many parents say things like this: "If that son-of-a-bitch doesn't shut his dog up, I'm gonna get out my rifle and shut it up for him!" or even unthinking, common things like: "Next time someone calls me a @#!%, I'm going to beat the sh*t out of 'em!" How about the parents that scream obscenities at their kids' sporting events? And then there are the ones who get into drunken brawls, or slap their spouse around, beat their kids, bully their neighbors.
Preventing children from thinking that killing is okay has to be a conscious choice on the part of parents, teachers, family, and the entire community. It doesn't just magically happen. Children have to be taught basic human connections, like how to empathize with other people. They have to be taught the social skills needed to deal with other people, like how to set boundaries for themselves, how to resolve conflicts effectively, how to interact with human beings in a healthy manner. I think that this starts with modeling that behavior in the way we treat the children, themselves.
But that's just my take.