Obviously, this guy doesn't care too much about what happens to his dogs...
With this type of person, only losing money might make them care. Call animal control every time you see the dogs running loose, even if they are not on your property, get a paper trail going. Animal control will pick them up, and where I live, there is a fine for dogs running loose, AND a fine the owner has to pay to release the animal from animal control. Sucks when you call and animal control shows up and can't find the dogs or they are already back in their yard, but if you can video them running around on your property it is usually enough proof. For some owners, picking the dog up from animal control is already so much of an inconvenience they just leave it there. I'm lucky where I live that animal control works with no-kill rescues in the area to attempt to re-home as many dogs as possible.
I completely understand that you would not shoot a dog. This should not be your problem to deal with in the first place. If you do feel comfortable talking to him again maybe politely remind him that it is YOUR property, and no amount of securing a coop against predators is enough to fend off a pack of 3 dogs forever. You cannot be home 24/7 on 'shoot the dogs in the a**' patrol' I am sure you have other stuff to do. You shouldn't have to spend money repairing your coop or replacing chickens because his dogs are doing property damage. Chickens are supposed to live in the country, I don't know what this guy is thinking acting like something will just kill them anyway so it is acceptable. You might also want to remind him that 'in the country' farmers would not think twice about killing a dog, even if it was their own, if it could not be trained to get along with the flock. If he doesn't respond well to that then tell him next time they set foot on your property they are dead. You don't have to do it, just make him believe you will. You should check, but in most places it is legal to defend livestock with lethal force. The only hangup is if you live in a town that has a ban on discharging a firearm within town limits.
Not to mention that otherwise 'friendly' dogs can sometimes act differently in packs, and a pack of untrained big dogs can be extremely dangerous to children as well as animals. Unlike wild animals, these animals are his RESPONSIBILITY and you are probably doing the whole community a favor by pushing the issue. I love dogs too, but it is certainly not your fault if they end up getting put down because this guy refuses to follow rules.
In the mean time, I would strongly suggest getting yourself a 'hot wire' or other electric fence/netting because it might be the only thing to give you some peace until this is resolved.