If you are looking for an actual list of banks that do not use chexsystems, here is such a list. I’ve compiled as accurate a list as possible, but keep in mind that the list is not 100% accurate. If you find contrary information, please leave such a comment. The bank list are listed by state.
Banks That Do Not Use ChexSystems (By Map) — Select by moving mouse over appropriate state and clicking
Banks That Do Not Use ChexSystems (By Table)