Pet Resume
Some people can be real sticklers about big dogs, but it's because they don't know any better. What I did, by recommendation from the ASPCA website, is put together a doggy resume for my 2, detailing their training history, temperment, health, activities, etc. so show a potential landlord/lady that I take responsibility for my dogs and they are well-cared for.
Responsible dog owners are good tenants. If you show them that, they are likely to "bend" their rules for you. This is more effective with privately owned rental property than big apartment complexes, obviously.
Another tip is not to tell them on the phone or via email that you have a large dog. Tell them in person after they get to know you and give them the resume to consider. I've found that in person, people are much more likely to consider a larger dog than they would over the phone. I might not try it on places that absolutely don't take animals, but if they state that small animals are OK, I think you have some room to negotiate.
I can't find the link to the doggy resume example on the ASPCA site, but here's a link to
Tess & Conner's Resume that you can use as a template. Note: I've removed all contact information.
Good luck!