Here in Colorado, I have tried the "no pets allowed" with one of my patio homes, to very disappointing results.
I had a difficult time getting it rented when I tried "no pets", even when I lowered the rent. So after a couple of months, I gave up, raised the asking rent back up plus some, allowed pets, and had it rented within days.
I do require a $500 pet fee, with $250 non-refundable for cleaning.
I do not place a size limit on the dog. I do prefer dogs to cats, and I've been lucky in that I've never experienced any issues with the couple of cats I have allowed. Anyone with any knowledge or experience with dogs will realize that large dogs are generally much more mellow and less troublesome. It's the small ones more likely to do damage - scratching doors, defecating in the house, barking, biting.
One of my tenants had a Great Dane, and it has been my best "tenant" so far! We have have breed restrictions in the city, so I don't really need to do anything about breeds.
I do my own screening of tenants, and I own dogs myself. So I feel that I can get a good feel for what type of pet-owner they are in just a casual conversation with them. I've never had any serious pet damage in my 17 years of owning rental properties. Call me lucky, but I think it's just good screening.
I think some states may be less pet-centric than here in Colorado, so it may be easier to get houses rented without allowing pets. But here in Colorado, I've learned my lesson.