First, to answer your question. Most landlords when they run a "rental history report" the main things they are actually doing is checking your credit and eviction history. In general, most people that have Decent credit are also decent tenants.
That said, if you have a list of happy satisfied previous landlords and steady income coming in, stress those two points to potential places and you should be able to find a landlord ready to work with you!
Its probably a good idea to run your own rental history report to see what landlords know about you. Check out myrentalhistoryreport.com to get one.
Also, check out a service like rentalkharma which lets you rebuild credit by paying your rent on time. Could be a great resource for you.
One way to be prepared is to know what your tenant screening report looks like ahead of time - you can go to MyScreeningReport.com and order your own tenant screening report (from a reputable screening company). This will allow you to see your credit report (with score) and background check - including rental & employment verifications, if you choose. This screening company allows you to share your report with multiple landlords through their secure site at no additional cost! This allows you to be in control and shows prospective landlords that you are serious about and prepared to go through the application process!
Visit http://www.themadsengroup.com to see if they have any rentals that might meet your needs.
Principal Broker / Owner
Hightower Homes LLC
Apartment Search by Cort can help you with your search. They keep track of most apartment community guidelines.