Or . . . reconsider taking your family with you. Okay, I understand that there are several and serious implications involved with moving your family out of the country; but how many kids, do you know, can speak more than one language (not counting their natural language or Spanish)? It could broaden your children's "cultural" horizons and their overall potential. (This little bit might help convince your wifey. . .

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If you were to consider taking your family, you would want to be careful about the place you choose to live. Dubai, from what I hear, is a pretty cool city. There are so many non-Arabs over there that another family of Americans isn't that odd. Abu Dhabi might be pushing it. And although I really like China (mainly because an American salary = $$$), it wouldn't be the best place to raise a family. Germany, France, Britain, Scandinavia, Guam, Korea, Thailand, Vietnam, French Polynesia, Malaysia, the possibilities are out there. The best thing would be to talk to your wife and see if she would even consider ex-patriotism. If not, well then you're stuck flying international for and American Carrier.
So back to your original question. I have been doing a ton of research over the past several months in order to satisfy my urge to fly ultra-long haul trips. Australia is not a place for American Pilots. New Zealand is a little bit more friendly and the person you would want to talk to is KoruPilot, of this very forum. There are several companies that hire American pilots, to name a few: Cathay Pacific, Dragon Air, Korean Air, Tiger Air, Air India, Air Philippines, Air Singapore, Eva Air, Japan Airlines, Emirates, Kuwait Airways (a nice airline to work for according to my dad who did so for several years), and Qatar. This is only a sampling, a thorough search would get you a few more, including some European ones.
Speaking of European airlines, that pilot market is highly competitive. So pretty much forget about most of the big names. Ryan Air is an option (if that suits your fancy.) If you can speak, read, and write German, checkout Lufthansa. SAS likes experienced pilots, but from what I hear, they prefer native Vikings.
There really is no special method for getting on with an International Carrier. Of course if you know somebody in the company, your chances have the potential to increase. Pretty much locate their web-site, send in your resume, and wait. Some (specifically Emirates) have infamous interview programs, others are more laid back. Most, from what I have heard, encourage you to bring your wife with you when you come for the interview. The last thing they want is a pilot who is having problems at home because his wife doesn't like the country. Lastly, just because they call you for an interview, doesn't mean you have to go. However, if you turn it down, make sure you won't ever want to work there in the future, as they tend to have long memories (specifically the Asians.)
"Honey, let's go on vacation. How does Hong Kong sound?"
One more thing, check out
www.pprune.org, as this site contains a wealth of international information.
Hope this helps.