I don't really have much advice, but I know it is possible! My downstairs neighbors have three large dogs and rent....I believe they found the place through Craig's List (that is how we found ours). I have one dog, but she is medium sized so that makes our renting life easier.

Also, having references from past landlords can help a lot and stating that right off the bat when e-mailing a landlord. And looking outside the city. Maybe Malden?