Philip Green - Sorry, I don't want to come across as a jerk but your dog does NOT clearly understand that he's not supposed to do that when you're not around. He may look and act like he does (all submissive and to our human eyes remorseful) but dogs are expert body language readers - you come out upset about the dead bird and he's reacting to your body language or previous negative associations (when there is a dead bird laying on the yard, my owner get mad at me). I promise. If he understood that there were negative consequences to something he did 2 hours ago that somehow you are aware of, well he'd be a superdog.
For both chicken killer dog owners it seems like you have to either make the chicken chasing/killing unrewarding/unpleasant - a la Walter's experienced suggestion or change the dog's mindset about them by making positive associations with being calm around the chickens. Being calm around chickens > chasing/killing chickens. How you do that depends on what your dog's personality and preferences. The dogs certainly have to be controlled completely around the chickens, rewarded for behaving well and put away for behaving badly (if not being with you is perceived negatively by your dog).
However, I think the key is that he didn't use an electric collar on any of the other dogs - the dog he talks about training was neglected and lacked a lot of foundation socialization I imagine.
So my advice would be to look at your relationship with your dog. Are you his wonderful, amazing human who is the best thing around (you have the dog's attention) or are you of the mindset that you have to be the "dog alpha pack leader" who the dog respects/fears (when you're around...)
I guess for severe
dog lovers, if you look into the world of positive reinforcement training, you'll find some answers and build a wonderful, trusting relationship with your dog. The solutions aren't 1/2 hour tv show fast but what do you expect when two conscious, unique, individuals of completely different species are trying to make each other understood.
Otherwise, the quick fix of baiting the chickens somehow so the dog experiences negative consequences for what they're doing while you're not around (very important) will most likely also work.
Regardless of which option works best for your situation, until you train your dogs, you need to manage the situation so they don't have access to chickens - the more they practice the behaviour, the harder it will be to eradicate.
