I have been wondering the same thing myself, only about the Gulf Coast. Our credit problems aren't unpaid credit cards, but hospital bills. I find it silly that someone can look at something such as multi thousand dollar hospital bills (yes, with insurance) & say you won't pay your rent. It's not like we can help it when a family member gets admitted to the hospital for a weeks time twice in 1 month, 3 times within a years time, being charged a $250.00 a day copay. Not to mention the multiple tests, procedures etc. As it is we pay 3-5 hundred a month for health coverage. Then you factor in all of the pay that was missed when in the hospital, it makes it nearly impossible to pay them off. Especially when you have to play catch up due to 4 weeks of less than half of your pay. I tell you what though, or rent wasn't late the entire time. Sorry for my rant, it's just frustrating & doesn't seem fair to be prejudged due to things that are beyond our control.
My best advice is to try to locate local renters. Usually they don't run such extensive credit checks, if any. Also, having a stable, solid rental history helps out. If your previous landlord can vouch that you've always been timely in paying rent, it helps. Some complexes don't run credit checks, but they also don't advertise that. So maybe working with an apartment finding service, or a real estate agent would be helpful?