Is it possible to have magento transfer payments directly to a bank account, without having to go through paypal or authorize.net or anything like that?
You can enable one of the offline payment options and provide them details. The payee will however need to log into their bank and do the actual transfer you cant setup an ecommerce system that will do that part.
I would suggest the check/money order option, edit the email template that is sent out on order completion to include your bank account details, or edit the success page to show that information to the user. (or both).