I agree with the part about Theoden being sad. I also found the parts about the Elves sad as well. I found the part when Sam thinks Frodo is dead sad too. And the part when Sam is being left behind, torn between his family and Frodo, as Ross said.
I find the song: 'Where now the horse and the rider?..." extremly sad, strange I know. I think it's just the concept of their culture fading away that I find such a loss.