So, you already got a lot of answers which tell you the situation.
Momentarily I work for a major European carrier.
As far as I have seen: It is not allowed to take anyone on the cockpit jumpseat. It is written in their Operation Manuals that it is forbidden because of several safety agreements after the 9/11: no-one on flight deck who is not in charge with any direct flight duties.
And the European cpts sometimes don't even allow relatives of the first officer (me in that case as my experiences) to sit in the cockpit when we fly to USA - because they don't want to get any penalties. Sometimes they are not so rigid when going other destinations than USA.
But you can have a stby ticket when there is any agreement with your company and the one you want to travel with. It would be quite cheap, and as far as I have seen, the crews don't refuse to take anyone on home travel with a stby tx as long as there are free seats. To get offered a cabin jump seat in case the a/c is booked out - just a matter of luck, I wouldn't count o it.
But anyway, what is the airport you want to go to USA from??
edit: sorry, I just read: you want to go to Asia. So, there shouldn't be any interferences, you would get any free seat with a stby ticket, and the aircrafts are not booked out. And the flight deck will invite you for some visits, I am sure, because they are curious to talk to other pilots and talk a bit about airline matters...
I think you want to go to Korea or China? Just get stby tickets or try as the other one told: Try it with some cargo airlines.