In addition to Suzanne's suggestion--which is the first thing you should do--ask the tenant for all locations he/she lived in for the past 5 years. And ask for contact information for the owners or landlords. Then contact each personally. Be careful that the tenant hasn't given you the name of a friend or accomplice. But that's pretty easy to detect.
Tip: To detect whether it's a friend or accomplice, here's what you do. Let's say the tenant says he/she paid $1,000 in rent. You call the reference and say, "I understand that [tenant] paid $1,100 a month in rent. Is that correct?" An accomplice--figuring that's what the would-be tenant said--will say that the number's correct. A real owner/landlord is very likely to say, "No. Actually, [tenant's] rent was $1,000."
Hope that helps.