There is a distinction between doing 3rd year core clerkships and 4th year electives.
If you want to do 3rd year cores, you will have a harder time setting these up on your own - especially if you do not have "connections", as you say. Ross, SGU, Sackler, a few other more reputable Carib schools, UAG (I believe), and one of the Filipino schools have programs to do core rotations in the U.S. Most are set-up in NY because these schools are "pre-certified" to by the NY state education board to receive this training full-time at U.S. teaching hospitals. Obviously, this is an advantage come residency time for students attending these schools because they can definitively say they've done all their clinical training in the U.S.
Most states restrict the amount of time that IM students can participate in pre-graduation (i.e., non-PGY) clerkships in the U.S. Additionally, many schools further restrict the amount of time even further (the most I've seen for visiting electives, with school approval, is 12 weeks).
Everything else that Dr. Cox says is right on the money. You can look on the AAMC website for a list of programs that accept students for extramural electives.
Good luck.
-Skip
MSIII RUSM