Violence in Games, do we need it?

My wife wanted to watch the movie Inkhart, a movie that never really got my attention. It is actually a good movie. There is suspense and life threatening danger, but no real violence, as in someone getting their head chopped off. (It is rated PG.)

It is a question that I keep stumbling over and over in the last time. I stumbled again over it as the credits rolled:

Is it possible to create a deep and thrilling action-adventure game without violence?

The reason why ask this question, is that film and books can create this sense of dead and thrill without actually getting into the act of killing. I am not an opponent of violence in games, it has it's place like in any other medium, but is it possible? It probably is but I can't figure out how this could be accomplished.

It seems to me that by taking the game mechanic of killing out of a game there not much left that it game like. One solution could be, as employed by Heavy Rain, to build a story driven game. But arguably this hardly qualifies as a game, since you are mostly sending your characters along a somewhat rigid path.