The following banks and credit unions offer excellent features and benefits. While we recommend them, they are not our top picks for the simple reason that they are not guaranteed banks. They have however helped a lot of people, and may be able to help you too.
Generations Credit Union
They do verify identity and run credit report but still may be able help you through one of the programs/accounts.
Woodforest National Bank
You must apply in person and they are friendly to those reported to ChexSystems.
Many people have been able to get accounts with TCF. They use ChexSystems but are extremely lenient, definitely worth a chance.