We've been autopay for as many bills as possible for years - several years ago instituted the "desk" credit card for making autopayments per a suggestion here. We also have our credit cards set to pull the statement balance each month, which has worked great with Fidelity, Chase, Discover, AmEx, PenFed, Citi and I'm sure some others.
Notably BofA, which was very tough to setup for an external bank card autopayment, bollixed our card autopay 7 months in a row, repeatedly pulling double payments from our external bank (very exciting when we had a $14k bill one month x 2) and charging and reversing late payment fees. Spent hours with them trying really hard to keep and use a BofA card that would pay 2.62% - finally admitted defeat and dumped BofA in favor of Fidelity and their 2% back card - and never an issue.
I'm hoping Navy FCU will soon allow their 2% card to be autopaid via external bank - I've had several encouraging messages from them to that effect and have nothing but praise for their responsiveness.
We do camp out on Quicken daily and keep very close track of the income and outgo, but autopay? dam betcha!