You can call local dog rescues and the humane society to see if they know of dog friendly housing. My local animal shelter keeps a list of dog friendly housing because moving to a place that doesn't allow pets is the number one reason for people to take their pets to the shelter. You should create a dogie resume for your dog showing his vet records and any training info, and references from his vet and trainer. It would help to put a CKG title on him.
We are renters-we've had the most luck with renting private homes found on craigslist but we have had to consider renting in neighborhoods we would have never looked in had we not had dogs. Good luck.