My prompt goes something like this.
"Hey so and so, I enjoyed having Daisy over this weekend. If you get a moment, would you mind leaving a review about your experience?" OR "I hope you were satisfied with Daisy's boarding experience this weekend. Would you mind leaving a review so others can read about it?"
Sometimes I even throw in that I'm trying to boost my search rank, but that's usually only with repeat client who haven't left a review in awhile. Use your best judgment on verbage with the type of person the owner is (sometimes it's not even worth it).
I won't usually follow up if they don't leave a review after my first prompt, unless it was an especially phenomenal or unique stay. Unique, for example: I walked this girl's dog at the Ritz-Carlton while she was in town. She had to put my name on her hotel room for me to get a key, which means a lot of trust, and I was extremely flexible with the dog's walk times as her conference times changed throughout the day. Despite being hired just for a walk, I played with the little guy in the hotel room sincce he was stuck in the bathroom, in addition to walking as long as he wanted. I definitely want her to leave me a review because I was awesome and it was a new type of service (from a hotel) for me.