I use the "debraining" or "pithing" techinique before cutting the throat. My chickens are put into the cone and their feet are held with a noose for added security. I then open their beak and stick a sharp boning knife up through the roof of their mouth into their brain and give it a twist. If done properly, it kills the brain instantly and the chicken goes completely limp. I then slit the throat. The heart is still beating so the bird bleeds out completely as if you hadn't first done the debraining technique, but I have the confort of knowing that as far as the brain is concerned, it is already dead. There is still a moment or two as the last blood is leaving the body that the chicken thrashes which I always find disturbing, but again, I have the comfort of knowing that it is just that wierd thing chickens do, even if their heads are completely removed.
If you decide to use this technique, look for info on it on the web. Find a picture of the chicken's anatomy so you know where the brain is located. You actually have to go up and backward, think between the eyes with the knife so that you get the brain rather than just damage the poor thing's beak without killing the brain. For me, this seemed the best combination of humane killing and least chance of the chicken damaging the meat running around without a head and spraying blood everywhere.