It is not the case that only a "few" US schools accept non-LCME students for 4th year electives. I had no problem finding schools, and big name ones as well, which received my application favorably.
Core rotations are those that US students tend to do during 3rd year: Surgery, Medicine, OB-Gyn, Peds and Psychiatry. Now most schools will NOT let visiting students do Core Rotations - those are reserved for their own students. An elective is pretty much anything outside of these cores and is generally done during the final year of medical school - it can be a subspecialty of one of the Cores (ie, Plastic Surgery, Cardiology, etc.) or it can be something like Radiology, Anesth, etc. These are generally available to visiting clerks although first choice again usually goes to home-school students.
To do a US elective at a school that does accept applications from international students, you must have liability/medical defense/malpractice insurance in effect in the US (check with your school), health insurance in effect in the US (consider travel insurance if you don't/can get this), good academic standing at your medical school, a clean bill of health (ie, must show that you've have Hep B series, TB testing, etc.) and submit an application. The schools will vary with regard to what else they need - ie, some will require a faculty sponsor which can be as easy as emailing the elective supervisor and asking.
Some US schools which DO take international clerks (off the top of my head; I know there are MANY MANY more):
UCLA
UCSF (while their web page states they don't take foreign students all of my classmates from Flinders in Oz have been accepted for electives)
Hopkins
UVA
Case Western
Baylor
UCSD
U of Louisville
Stanford (although charges going rate for tuition I believe)
Harvard (ditto)
etc.
Most schools have web sites which detail their policy. Its simply a matter of looking at schools and asking for/downloading the application.