While I agree with Bruce's approach of picking a company that apparently displays a record of success, remember that companies are run by people and people have personalities.
Make sure you pick a personality that you like. It's not just about price or ability, you have to work with the company and understand their approach to management, their policies and match that to what you want.
A property management company that can't retain tenants in a property might have a problem, or it might be a problem with the property. Ask about their tenant retention rate as well as their client retention rate.
Some management companies focus so much on volume that the out of the ordinary situations can result in stress. These days collection has reared its ugly head as an on-going problem and choosing among applicants is not as automated as some websites would have you believe.