I'd have to say Fellowship of the Ring made me more sad than Return of the King. They are darn close though. I don't really cry but I do tear up. If I do cry, that's when you know I was
really upset about it.
In Fellowship of the Ring, I always tear up when Gandalf falls in Moria. Seeing the sadness of the hobbits just makes it worse. Sam almost drowning gets me too. The one that always gets me and actually makes me "cry" is Boromir's death. It makes me upset just thinking about it.

I'm typically a mess for a while after I see it.
In Return of the Ring, I got teary when Faramir said, "You wish now that our places had been exchanged..." (I have a place in my heart for the brothers, I love them.) Now matter what Faramir does it never pleases Denethor, it is clearly shown by that scene and it makes me sad. Arwen's vision mad me a little sad to.