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.
They do not use ChexSystems and they do not run a credit report at all.
Library of Congress Federal Credit Union
They do not run any type of credit reports or check reporting systems when you open a checking or savings account with them.