Here's what usually worked for me (having been on both sides of the interviewing table):
1. Be yourself - don't pretend to be someone you are not. If you are typically shy/reserved, don't try to be over gregarious. There is great value in inward-focused people (I am one) and I admire those who tend to be themselves when pushed into a stressful situation like an interview. This alone with "connect" you to your interviewer at a very elemental, human level.
2. Take the time to listen to the question and understand it. Ask questions if ANYthing isn't clear. We really appreciate you taking the time and effort to understand the case.
3. This won't work everytime (and mckinsey didn't reveal names before interviews so I couldn't google them up), but observe something interesting about the interviewee and, if appropriate, find a thread of connection there. I got lucky - my interview happened in the interviewee's office and he had photos of his mountaineering lifestyle in the room. It was easy from there on. It could be anything; their jacket, tie, pen, whatever. Their accent is the easiest to go for ("oh, may I ask where you are from: I hear a bit of kiwi.. and have lots of friends in queenstown; been dying to visit..etc etc).