Speaking as a former small-time landlord, I don't think I'd rent to someone who was recently foreclosed on. If they couldn't pay their mortgage, I have no reason to believe they could pay the rent, even if it is a lower amount. People who have been foreclosed on often are unemployed, and often have big credit card debt, gambling problems, huge medical/childcare expenses, etc. which would all get in the way of paying rent on time and in full.
Would you rent to someone that was foreclosed on who would give you six months or the entire 12 months rent upfront?
With non-recourse loans, you can't easily make assumptions about someone's finances. If someone just wanted a quick flip and made a bad investment and is now underwater $200,000 or $300,000, the logical thing to do is to cut your losses and tell the bank to shove it.
Let's look at this example. Someone buys a house in CA for $750,000 at the height of the boom in 2005. They put 5% or $37,500 down with monthly payments of $4,000.
The house is now worth $450,000 and they are still on the hook for $48,000 a year in payments.
They can get the same house now renting for half as much.
It's very easy to see why this person would tell the bank to shove it.