Cathay Pacific Airline?
The Taiwanese airlines, China Airlines (CI) and Eva Air (BR) often have very good deals on tickets when compared with United, Singapore Airlines, Cathay Pacific when flying across the Pacific, especially for business class. CI's Dynasty business class is better than the old EVA's Super Business Class in terms of seat size and pitch, and about on par with Eva's Evergreen Laurel or Premium Laurel, but Eva has the Evergreen Deluxe, which is a kind of super/premium economy class which is great deal for the money.
However, if you aren't terminating in Taiwan, you do have to make a connection in Taipei's airport, which isn't all that interesting, though perfectly functional. Also, if you are accumulating frequent flyer points, these carriers FF programs don't really transfer well to other programs, which could be a problem.
Cathay Pacific (CX) is the Hong Kong based carrier, which has excellent service. HK Airport is well situated for connections onto Asia - they have nonstops from San Francisco, Los Angeles, New York, Vancouver. They are a One World alliance member, with BA, AA and JL so you can earn points on these other carrier programs as well.
Another airline with world class service worth considering is Singapore Airlines - they fly to Singapore from.San Francisco via Hong Kong or Seoul, Los Angeles via Taipei or Tokyo, and New York via Amsterdam or Frankfurt (I think). They have a non-stop flight from LAX to SIN on the A340, and also am the launch customer of the biggest jumbo jet to be built, the A380. They are a member of the Star Alliance (with United, Air Canada, ANA, Lufthansa and others) so accruing mileage with them is convenient.
I think your decision to take a flight on a certain airline will boil down to a combination of price, flight scheduling, service and frequent flyer affiliation. Good luck!
I live in Houston and have used Continental for over 40 flights to Narita..not boasting,,but job. I am happy with the service and the 777.
Cathay Pacific. Good service, good food, and big airplanes.
Of the three you proposed I would choose Cathay for economy class.
Eva has a deluxe economy which is business class seats with economy class meals. It's really nice for the long flights but you will transient Taipei which is a boring Airport.
Off the list I would advise Continental, out of Houston, if you're flying from the South West. It's a comfortable non-stop to Tokyo on a 777.
From LA or New York Singapore Airlines has non-stop flights to Singapore that are very nice if you're going to South East Asia. But if you fly SQ their mileage program is the pits. Join the frequent flier program of one of the other Star Alliance carriers and log your SQ miles with them.
If you plan to fly around Asia use Jet Star for the best deals but try to book in advance.
Try the Emirates Airlines via Dubai - UAE. They are realy good.