I would add another alternative not mentioned yet: Aeroflot via Moscow: you could leave the USA from SFO, LAX, SEA, JFK, MIA, WAS, ANC,... (maybe flying B772 or B763, or, if you appreciate and have the opportunity, a Russian-made plane) and make a connection to Mongolia, flying probably a Tu-154 or a IL-86.
Meanwhile, one thing is sure: whatever way the traveller choose, it will be an adventure! (man, I'm from Brazil and I don't have ANY idea of how Mongolia looks like!)
TP343, São Paulo, Brazil.
p.s. the site B-707 mentioned is: http://www.miat.com.mn