A real easy fix is to call the cops or sherif, in most cases they will not know which neighbor called. Most cities have a law on noise problems like this...
Bingo. Doing harm to the dog(s) is not only a mean thing to do, it is a very likely way of getting yourself in real legal trouble.
It's always best to just be neighborly and just ask them nicely to please quiet their dogs. If that fails, call the police or the sheriff if you're not in a town. Not 911, but whatever the local non-emergency number is. In my neck of the woods, they will warn the offending party once or twice and then they'll start writing tickets.
Edit: I just saw your post that your city requires a bit more intervention from you. That's unfortunate, but understandable. I'd just start by trying to talk to the neighbor, if you can.
