Last year I opened a chequing account at the Citibank branch in Niagara Falls, NY and had an AAdvantage Debit MasterCard linked to it. I got 1 AAdvantage mile/$ spent on purchases which wasn't bad. But then I stopped flying AA and got tired of using the account so I closed it. Other banks that offer similar accounts are:
Bank of America - US/HP/AS
US Bank - NW
Chase Manhattan - CO
KeyBank - CO