Well, to me one of the saddest moments was when Gandalf fell in the mines of Moria. I read The Hobbit first, and then LoTR so I had a soft spot for the old greybeard already.... and it was so sad (and tragic) that he fell to protect the ones he cared about.
Of course when we find out he wasn't dead after all, I was really happy.
Another really tense moment for me was when the orcs take Frodo away. Perhaps it was not tragic in the same way Aragorn's passing away may have been, but I remember finishing The Two Towers and wanting to run to the bookstore to get The Return of The King... even though it was a weekday, close to 10 PM
