The process started with a quick phone screen with an HR person. The second interaction was a phone interview with a lead product manager. The interview consisted of some standard PM questions regarding metrics and feature development. The third step was a set of 4 1:1 interviews at the office. Unfortunately, since the first interviewer was late, the whole process ended up being a bit rushed. Questions ranged from personality questions, situational questions, my thoughts/impressions of the industry, and also direct questions related to specific issues LinkedIn was encountering at the time.
If you are interviewing as a PM, be sure to know the site/product really well and be prepared to offer thoughtful suggestions on improvements. For any suggestions, you'd likely be asked how you would implement.