I have been bitten by a pack of dogs when on the bike. My answer is pretty simple: bear spray.
I am also pretty shocked by replies in this thread. Are you people really genuinely scared of "rednecks" taking revenge? Geez. You make US sound like some 3rd world country. At least from EU perspective.
Absolutely! And we should be scared! Having lived overseas I can understand your disbelief with some of the replies. However parts of the US are very third-world like. There is significant poverty in many parts of the country (the rust belt, bible belt, parts of Appalacia to name a few). Even the rural areas where I live have much economic depression. I haven't dealt with the southern rednecks in rural North Carolina but know the stories. Our rednecks here in Vermont are pretty mean and scary too. I could write a book about my stories of being bullied by rednecks and their dogs while running and cycling. It's pretty bad. And they are vindictive, nasty people. One thing to remember is that we have very lax gun laws in the US so many of these rednecks are fully armed and have an assortment of assault weapons.
I'm torn on what advice to give to the OP. Part of me wants to stand up to these backward redneck f*&Ktards. But it might not be worth it in the end. If the problem isn't solved after calling animal control, I personally would find another route. It sucks! I'm with you!