Renting out: questionable history, please advise

[Scorpio Johnson]

We have an offer to rent out our property but the prospective tenants have some questionable history.  Low credit scores (low 500), unpaid medical bills, a few collection agency accounts, supposedly never late on rent.  One of the tenants has a misdemeanor (Resisting Officer Without Violence (Resisting Arrest)), found guilty.

We've rejected the offer because we were pretty sure this is not our cup of tea, but they came back asking if we would consider if they would pay 6 or 12 months up front. This is a brand new house and we'd like to find easy renters. Having an offer to pay 12 months up front is tempting, but what kind of issues could we expect down the road? This is our first renting experience and we would appreciate to get some pointers.  Any scenarios that we should watch out for?