Our dean specifically said not to do this. Personally, I think it looks kinda schmuck-ish
You think it makes you look like a schmuck to have business cards as a law student? I'm guessing you didn't spend long in the business world between undergrad and law school.
Let me explain, people keep track of a card much better than some scrap of paper that's scribbled on. No student is dumb enough to just hand their card to a potential employer without being asked. You only give it to them when you have a pretty good reason. I have had many attorneys explicitly ask me for my card, either because they wanted to contact me or they were passing it on to someone else; here, a card tends to make you look more like a professional, whereas a scribbled on piece of paper tends to make you look more like a student. I, as well as most of the attorneys you will network with, keep a binder specifically for the business cards (not random paper) of contacts, jotting down little things on them about the connection and the person to aid the memory.
In sum, utility is why I have them, not as some weird unearned symbol of power or whatever you may take them as. Law school is a professional school, and business cards are just tools of the trade.