DillyDally-
We own our home, but when we rented with dogs, I posted an ad to Craigslist (like KSL classifieds) and I said "this is who I am, this is my family, these are our dogs" and what we were looking to rent. Of course, we aren't typical animal owners, and we keep show dogs, so that helped. But don't be afraid to put up an ad and blow your own horn, so to speak. Let them know you aren't into hooliganisms (don't think that's a word!) and that you keep good care of your animals and don't let them run amok. We had several people begging us to rent their homes.Hope this helps!
Thats how we got into our current place, which we LOVE. I used to show dogs, so our pups are very well trained. I will just have to try that again and hopefully it works out just as well!