American Express really does monitor for suspicious charges
Yesterday I got an email that claimed to be from American Express saying they'd noticed a suspicious charge. I looked at the raw message and couldn't find anything wrong with it, but being careful I decided I'd wait and call the number on the back of my credit card from work this morning rather than call the number given in the email. I just made the call and discovered there were three suspect charges on my card. American Express declined all three charges so there is nothing I need do now except wait for my new card to arrive. And then contact all the merchants who make automatic charges to my card and give them the new number.
I have no idea how my card number was obtained by thieves but I am very glad American Express is paying attention.