I traveled to Costa Rica with Amerispan during the summer after my first year of med school. It was a wonderful experience.
Pretty soon, I'll be leaving for Ecuador with Child Family Health International. I'll be doing a community medicine rotation, but they have rotations available in tropical dz, surgery, and something else. The organization also has rotations in Mexico, South Africa, and India. They're all pricey though--the Mexico one is the cheapest, and teh India one is very expensive
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You should check the AMSA website. A couple years ago, I contacted people in Africa, Australia, Thailand, Nepal, S. America, and many others. I heard back from most of them too.
And it IS possible to do electives abroad during residency, depending on which specialty you apply for. I plan on doing this in residency, and only ranked programs that would allow their residents to travel abroad.
Hope that helps. Let me know if you have other q's.