[villagetinker]

There have been some posts on this in the past, but IMHO, I do not believe there is any thick shrubs or ornamental grasses that will provide a suitable way to keep a pet in your yard. There are several that can be used for privacy, STAY AWAY FROM BAMBOO. Remember you need to keep your pet in and predators out. At a minimum chicken wire or similar may work if properly installed with the plantings, however this will not prevent the various flying predators and may not stop some of the others.
You may also need ARC approval for your project and depending on your lot there may be various setbacks or right of ways where you cannot plant. We have small pets, they never go outside, they are allowed in the birdcage, and it has pet resistant screening.