The credit card companies have entire pieces of their businesses devoted to credit card fraud - tracking, preventing, and resolving. As good a job as they do, it never seems like it's enough, though!
We've got 2 credit cards, with Citi. Each and every time we go out of town on a trip, we call their 800 number to notify them that we'll be traveling. They want to know the start and end dates as well as the states where we'll be using the card. If we don't call them in advance regarding our plans, they will (and
HAVE) unceremoniously shut our card off when they start seeing out of state purchases for things like gasoline, etc. which indicates local use of the card there. Arguable downside of notifying them of our trip is the potential that some unscrupulous person on the other end of the phone could use the info to their advantage - and we could come home to find the place pilfered. (But good luck getting past the security system!

) It's a risk we're willing to take, though. Helps us feel good that Citi's paying a good amount of attention to preventing fraud.
Closing point: If you do have a number of auto payments go through your credit card account, consider getting a different card/account that you use for most of your other purchases. That way if your frequently used card gets compromised, your auto payments and the other card aren't affected. Just a thought.
Bill