I definitely remember my great grandmother just picking them up and whirling them around her head to kill them. Worked too. Practice, I guess.
I, too, read in a book that you could kill birds by breaking their necks. I had a one week old chick that I couldn't seem to help get any better so I thought it was time to put it down. I figured I could just break it's neck like the book said (I couldn't find my hatchet). It didn't work and the porr thing was chirping at me. I had felt the bones crack but it didn't die. I was squawling as I tried to finish the job. Haven't tried since. Fortunately, I've haven't needed to either.
Cresty