This only works with civilized neighbors. It's a judgement call. The problem is that if you try talking first, and they're not civilized people, they'll know who's responsible for what comes next. People who aren't civilized tend to get angry and confrontational when they have to live by civilized people rules.

If the diplomatic approach fails, take their dogs shit and smear it on their front doorknob. Follow up with the handles to their car doors.

Talking usually only works when both parties care. The dog owner sounds like he doesn't give a shit.

We have this problem in our area - to the point that the "domesticated pets" band together into packs and do great damage to pets and property. Laws are plain and known by pet owners, and we've found that talking with people who don't give a damn about their neighbors aren't going to see that they have a problem. If anything, expect random dog shit to be thrown into your yard after speaking with dog owning yahoos.

We now have a standard escalation of response: 1. Call animal control. Most areas have significant fines for allowing a dog to run loose - often compounded with other fines if the animal hasn't been properly maintained with vaccines.
2. Shoot repeat offending animal with pellet gun. It's not lethal, and they often learn to avoid the area after being hit once. 3. Shoot animal with subsonic rifle and take carcass to dump.

Dog owners who imagine that their beloved Fido won't opportunistically kill neighborhood cats and smaller leashed dogs are idiots who deserve losing their "pet". (A pet is cared for - allowing a dog to run loose is the opposite of care.)