To get on with LH is pretty difficult! Mostly they only hire the pilots who went through the "Lufthansa Training Program"! There is a possibility to be hired as a "ready entry".
Check out here Sorry, it is German but I couldn't find it in English! I'm not sure if they hire Americans. You defenitly got to have a permission to live and work in Germany.
I hope that helps a bit. I don't know about AF, BA and KLM though...
Just as information: I am US-citizen living in Germany. I have the German citizenship too so I wouldn't have the problem of being able to work here. I want to apply for the "Lufthansa Training Program" when I am finshed with school. If that doesn't work out, I will go back home to the USA and try it there... Lufthansa is defenitly a great place to work at though as far as I know. I know some LH-pilots and they say it is great! Good airplanes (The fleet is only about 7 years old in average and is even going to be partly renewed in a couple of years), nice employees (I can confirm that one myself) and decent pay!