Good job keeping your rent history clean. Any prospective landlord will be encouraged to see this. As others have said, each landlord sets their own standards for what they deem acceptable credit. There are many renters out there with difficulties in their past, so you should be able to find some options. Your credit is one area a landlord is allowed to use to be picky, so do what you can to clean it up. Credit repair/rehab is not as logical as it may seem. Do some research as to the best steps and apply them to your situation. Once you get your past credit issues resolved along with a clean rental history, you may want to look into qualifying for a mortgage, but first things first.
If they do run a credit check, it will also depend on how they assess the data. Give the same credit report to two different landlords and you could easily get two different decisions.