I don't usually cry for either books or movies, but with Tolkien it is an exception...
I always cry when Boromir dies... and the Lament that follows, and when Gandalf falls.
I can't help but cry when Éomer thinks Éowyn to be dead.
When Faramir rides to Osgiliath and (in the movie) Pippin sings his song... the combination of the two... oh man........
At the end of Return of the King, at the Grey Havens... it's the passing of not only Frodo, Gandalf and Bilbo, but of the Elves as well.... and Elrond will ne'er see his daughter again... that breaks my heart.
In the movies, I cried at Aragorn's rousing speech at the Black Gate... don't know why, but that type of thing has always moved me... kind of like hearing the Star Spangled Banner blaring really loud.
(yeah I'm American!)
Also when Gandalf calls Shadowfax and he comes onscreen for the first time... and the charge of the Rohirrim. (not from sorrow do those tears glisten in my eyes)
Certain parts of the Silmarillion also brought tears to my eyes.... namely when Glorfindel falls fighting the Balrog (Alas! 'Tis Glorfindel and the Balrog), when Finrod Felagund gave his life to save Beren's, when Huan died, and when Beren passed into the Halls of Mandos.
-Elrowen Tinúviel