I've broken the necks of songbirds who were attacked by dogs and it's just horrifying. I might have to put on my big girl panties and do it, though.
You can also kill a chicken by cutting off the head.
Some people use loppers (things like scissors with extra-long handles for cutting tree branches).
Some people use a hatchet, machete, or ax with a chopping block.
I prefer to cut the head off, because it's easy to check whether I did it right: the head is completely separate from the body, so it must be dead, even if the body is still moving around.