I am surprised that the identity fraud people are advising people to destroy old cards.
As an engineer, I know that there is no personal information... all that is encoded on the card is information on the bank and the credit card number. I think the fear is that somehow old credit card numbers are useful to thieves.
I know how the chips work. It is standard encryption technology that we use all the time. Again, there is no personal information stored (and no reason to store it).
There are internet sites recommending that you cut the card into four pieces making sure that each piece contains a different section of the number and then putting each part in a different garbage can. This sounds awfully excessive to me, but they are protecting the now expired number, not any personal information.
I have lost a couple of cards... including one that a thief tried to use before I realized it was gone. The fraud prevention people picked it up, I canceled the card. No harm done. How many people haven't ever misplaced a credit card.
I haven't heard any stories of people being hurt by not cutting up old cards. I suppose if the identity fraud people recommended it, then people should do it. Cutting the chips in half will do the trick.