If you are looking on craigslist, you can always sort for rentals that allow pets. A landlord who allows pets really shouldn't be able to discriminate based on the breed of dog. I have found that a problem some dog owners have is that they have weight restrictions for the pets they allow on their properties and that makes it hard for a large dog owner like yourself.
Have you thought of possibly buying a place? There are many options for financing a home and perhaps you would qualify for something where your mortgage would be about the same as your rent. Contact me and we can discuss the options you may have.
CA DRE License #: 01904666