Of the four dogs we have/had -
Newfie/Golden Ret. mix - very remorseful chicken killer - wanted to play with them, but they just didn't play for long. He killed 5 of our chickens and always brought them to me and would lay them at my feet with a remorseful look, crying and gently nudging them as if to say, "make her better, mom."
Lab, Beagle, and Pit Bull mix - Not really sure what to call them - they don't harm the chickens at all, but they used to sneak under the chicken wire into the run (before we finally fixed it) to eat the scraps that were throwen out for the chickens. I would be looking for one of the dogs and go over to the chicken area, and there s/he would be, laying patiently with the chickens waiting to be let out. Never a chicken harmed.
I think it truly depends on the particular dog. While training can help, some dogs don't need it and some, no matter what you do, can't seem to help themselves. Now that I have a huge fenced area for the dogs, I just choose to keep my dogs away from the chickens as much as possible if they're not supervised.