Tech writer recommends physically destroying old computer hard drives, noting that deletion of files still leaves the data on the disk which can be retrieved by ID thieves.
I know of one couple who had an old, unused computer stolen, and the thief used stored credit card data from online banking and shopping to steal their identity and make numerous online purchases with their credit cards. While it's possible to eventually determine the location the frauds are occurring from, it's generally impossible to prove who is doing it or prosecute them.