I own rental properties and I never rent to anyone without doing a credit check and background/criminal check. I have, however, rented to people with credit problems. In those cases, they put down a larger deposit and signed a lease stating that eviction will start sooner than our usual lease states. Different states have different laws, of course. Sometimes, it's county-by-county.
I am constantly surprised at the fact that so few tenants actually want to know what the contract says before they sign it. If you find a landlord who rents a house without doing a credit check, be sure you understand that contract!