Fortunately, there's an easy solution: cancel your account. Yep, just push that big red button (well, actually, it's a smallish grey button, but you get the idea). You'll still get to play until the time you've paid for runs out. And since Blizzard retains your character and account info indefinitely, you don't have to worry about your characters getting deleted. At the end of the time you've paid for, when you try to log in, you'll get a notice that your pre-paid time has been used up, at which point you can go on the web site and add whatever payment method you like.
I've used this method many times myself. It's only a few extra clicks, and if you want to have more control over how you get billed, it's definitely worth considering, even if it does make the peons cry.