Switching banks frequently can mess with your credit score, so I'd try not to do that too much (especially if you're considering buying a home/car/etc.).
They do report to a separate group of risk-scoring companies, but this is mostly to gate fraudsters. Frequently switching checking/savings accounts will have no bearing on your ability to make large purchases on credit. If you happen to close credit cards frequently enough, regardless of whether they are issued by banks, your score could be impacted by reducing your average account age.
I had this occur with Capital One and a scam fax-to-email company who refused to accept a trial cancellation. Even after lodging a dispute the scammers could still bill more charges, and I'd have to dispute each one until the bank made a ruling.