I would not recommend paying any bills via direct debit to your checking account. If that information is compromised, anyone possessing that information can then access all the funds in the account. There is no limit to liability as there is with a credit card in the case of fraud.
We do have our bank, Wells Fargo, pay bills on our behalf by setting up the payment via Quicken. Most recurring charges are automatically applied to a credit card and the credit card is paid from WF, again, via Quicken.