I called our credit card company and disputed the charges. They, naturally enough, are canceling the card and issuing us new ones.
Today I did a bit of further reading, and saw that my next step should be to request copies of our credit reports. I got in touch with Transunion and got that report, but that took nearly an hour and I still have two agencies left to deal with.
From what I've seen on the web, the theft does not appear to be associated with a particular company, the type of credit card, the victims' buying habits, or even a geographic region. Cardholders here in the States, in Australia, and in the UK have been hit. An article on MSNBC indicates that "federal authorities" have begun investigating these charges, but does not name the agency allegedly conducting the investigation.