Well, I cried a lot in LotR, but much less in the Hobbit because I wasn't really invested in the movie characters (totally invested in the book ones BTW), except for Bilbo.
For all three Hobbit movies, I think it's safe to say I cried every time Bilbo did, just because Bilbo.
For LotR, there are so many....
FotR:
After Arwen makes it across the ford and Frodo is on the threshold of death.
When Gandalf dies.
Breaking of the Fellowship, just the entire sequence.
Also Boromir's death.
TTT:
Faramir's flashback when he sees Boromir's boat.
(wow, I thought there would be more here.)
RotK:
After Frodo gets stung by Shelob and Sam cries.
Theoden's death
Eomer finding out Eowyn has been wounded
Denethor saying, "Yes, I wish that." AUUUUUUUUGH!!!!!
In Houses of Healing I didn't cry, but my expression was the same as Eomer's.
The entire last half hour.