For Dead Money, http://www.youtube.com/watch?v=VJVREj11lcA by The Alan Parsons Project, as Sinclair mentions the Poe story directly as he plans his revenge.
For Honest Hearts, http://www.youtube.com/watch?v=eJlN9jdQFSc&ob=av2n by Johnny Cash sums up Joshua Graham's feelings towards the White Legs. "I am the right hand of the Lord, and the instrument of his vengeance." "I don't enjoy killing, but when done righteously it's a chore, like any other."
For Old World Blues I think http://www.youtube.com/watch?v=suazwrc2RPU by the Dandy Warhols is pretty perfect, perhaps http://www.youtube.com/watch?v=V83JR2IoI8k&ob=av2e by Thomas Dolby for a female Courier, or Doctor Dala.
Lonesome Road, I'm not sure - need more on Ulysses before I can think of something that would fit.