And they might dig (mine would if I had a yard) and make it difficult to keep a lawn green because of urine, and the owner might not be great about cleaning up the poo, or might think just mowing it is a great idea, which of course, stinks to high heaven and it's not the greatest fertilizer for the lawn, either...So, even great dogs and great dog owners will do more damage than a tenant without one.Is that worth more rent?  

They can dig and rings of dead grass appear where they urinate.

Sounds like mold or dog urine in places you can't see.

You can require ALL tenants to exercise responsible animal ownership, protect the property, minimize chance of harm to other people, and maintain acceptable noise levels that do not interfere with neighbors' enjoyment of their own property.A well-written animal clause will require the tenant to:Provide proof of current shots, if applicableProvide proof of flea control medication, if applicableDisclose the name and contact information of the veterinarian caring for the animal, if applicable (hamsters, for example, probably do not have veterinarians)Keep dogs in a crate or on a leash at all times when outside the dwellingUse a pooper-scooper, whether on the leased premises or another's propertyVary locations of dog urination, so as not to burn shrubs or lawnEmpty cat litter boxes daily into outside receptaclesControl  excessive barking that disturbs the neighborsAllow monthly inspections for pet damageImmediately clean any pet discharges and immediately report any damageAssist with Twice monthly HVAC filter changes for long-haired animalsKeep pet food supplies in pest-proof containersI think that if you approach the problem in this manner, you will be more comfortable complying with Fair Housing laws.

And their bedding with urine and feces will spill onto the floor/carpet.  

I have a home the unauthorized puppy urinated through the sub-floor.  

I have heard of some really great products out there that are made specifically to rid of cat urine odor.

I did laminate in bedrooms in one of my rentals only to have to rip them up because of pet urine in one room and another time there was a leak from the adjoining bathroom.  

And, they must have thought that going upstairs or outside to urinate was "too inconvenient" for them, because 8 bottles of uncapped urine was found in the back storage room.In the actual upper unit itself, the carpeting was pretty stained up and there was ground up food in it. 

indeed around here, many are getting deported for even 'minor felonies' as having been cited for urinated in an alleyway.Police America!