Well, for one I have to spend my time to submit a fraud report to my bank. If using my debit card, the money is gone until the fraud is confirmed. Second, I have to wait for a new card to arrive in the mail, then try to remember who I have set up on automatic payments using my old card. Call each one of them or visit their website to enter in the new numbers. The ones that I forget will possibly result in account suspensions, etc, until after the new number is entered. Fees may be charged, which most of the time will be waived but that again takes more time to deal with.
The credit card companies need to fix this, and chip/pin is not the answer. It should solve retail store card theft, but as online purchasing becomes more and more popular, chip/pin will do nothing to combat it. We need a rotating pin device, similar to PayPal and World of Warcraft uses, and tie that number to the authorization. That number/pin combo would be useless for future transactions other than follow-on transactions to/from the same merchant for subscription or refund purposes. That way when a card number is compromised it is useless since the attacker won't be trying to get more money for the original merchant. Instead the card issuers just tout "$0 fraud liability!!!11!!!1!" to the consumers and pass the buck off to the merchants. Chargback fees from merchants are a profit center for card issuers, so why would they want to fix the problem?