I do. I advise my landlord clients to do a three pronged background check, credit history, eviction history and criminal history. It only costs $25 per person (I can give you the vendor I use if you contact me here on Trulia) and I find that anyone who cannot successfully pass the background check just goes away! I also get employment/income verification so we know they can afford to pay the rent, and references.
My landlords are very happy! I have rarely had a 'bad apple' get past all of that, not that it doesn't happen, it does, but it's rare.