Keep in mind the purpose of a business card; it's to give someone your contact information so that later when they want to get ahold of you they can reference it.
Programs, rotation sites, PDs, etc, are never going to call you as a medical student. It's incumbent on you to reach out and contact them for everything you need. So they would never need these cards. On the other hand you should be collecting as many contact cards as you can from these people so you have them on hand when it comes time for interviews, etc.
[edit... clearly I need to be more careful in reading the posts in the middle of the thread... jlaw said pretty much this exact same thing...]
When I was an attending (before deciding to go back to fellowship) and medstudents gave me business cards, I'd hold on to them until I passed the next trashcan. Eventually, I just started telling the students to hold on to them.