When the Wings get knocked out, I still watch playoff hockey right on through to the Stanley cup finals, with the exception of some of the games with 10:30 starting times. I just have a hard time staying up until one in the morning for a Kings versus Ducks game.
I always root for whoever knocked the Wings out, for instance, the Wings pushed the Hawks to a seventh game last year and lost in overtime, a great series and when the Hawks won I could justify in my mind saying the Wings were the second best team in hockey because they were the only team that actually pushed the Hawks in the post season (my rules, I make them up).
I watch to root against the teams I do not like, the Canucks for instance, not much gives me the pleasure of watching them doing the handshake with their heads down after getting knocked out in four games by the Sharks or Hawks. A few others.
I always pick a team to root for in the other conference, last year I chose the Senators as my team, and the spanking of the Canadiens in the first round including a line brawl in the playoffs of all things was a delight to watch. When they lost to the Penguins it hurt almost as much as when the Wings lost.
So yeah, I watch, I hate when the Cup involves two teams I do not like, the Bruins and Canucks in 2010 comes to mind, I had no vested interest and didn't like either club a single bit. Equally, I have a hard time when two teams I have enjoyed in the playoffs meet in the cup final, such as when the Flyers played the Hawks. After three rounds of rooting for two teams it is hard to get a good hate on for one of them.
The Penguins are a team I love to hate. I was shocked when they fell apart so unprofessionally to the Bruins in4 straight last year. Shocked but not saddened. Not saddened at all. I respect Crosby and even love to watch him play, it might be a bit harder disliking them now that Matt Cooke and Kennedy have moved on.
Sorry, a bit rambly and off the beam, but the question is do you watch teams other than your favorite in the post season?