Umm, that's gonna be a lot of pain for you to convert from FAA to JAA unless you have about 3000 jet hours, multicrew and least 500 jet PIC on your license. JAA makes it really hard, means you sit all 14 exams.
Better you do your flight training at a JAA place in the US (best bet would be somebody with in house examiner).
A license transfer from JAA to EASA is coming up during the next 2 years.
If you wanna get cost effective, you might want to look at a JAA training place in Slovenija. They provide good training at reasonable prices and the weather is very nice from spring to fall at the adria coast.
Cheers, fluf