Do your homework first:


Obtain a list of programs on FREIDA on the aamc website. Click on individual programs for the email ids if the program coordinators/ directors and the websites of the universities or hospitals.
Check the websites for information about observership/ externships/ short term research projects.
note down the email ids and contact numbers of the coordinators.
prepare your resume/ cv in the american style. the format is available on the usmleweb website.
now write to the PC/PD to enquire about the observership/ externships.

What are the things to ask:

Formally introduce yourself. Mention any exams taken or pending results. Express a desire to match in their program and why you prefer that program over others...a nd that you wish to get some experience before applying for a residency spot.
Also ask if one is required to pay a fee for the same.
What are the requirements to apply for the position? Steps cleared? malpractice insurance? Favourable visa? Fresh / old graduate?
Do they offer / suggest accomodation?
Is their institution IMG friendly?
If you are interested in externship, ask about the visa requirement...whether they sponsor any?

When to apply :

The best period is April- June. Most programs are accomodating of observers and externs before the applications for the residency pile in. As the number of applications are increasing every year, the possiblity of getting US clinical experience during the interview season ( October to February) will be meagre. unless pre- arranged.

Wait:

You have taken the first step. Now its their turn to respond. Wait for 2 weeks. Resend the email..Wait another week for the reply. If there is still no reply. now is the time to call. Do not leave voicemails.. your call will not be returned.

Which places to consider:

IMG friendly program with IMG Residents in the hospital is the safest bet.
Some hospitals like Medical University of South Carolina (MUSC) offer observerships to candidates in IM/ PEds/ Ob Gyn.. but they hesitate to offer interviews / residency spots/ good recommendations to the observers.

Some institutes:

Chandler Medical center, Lexington, KY
Cleveland Clinic, FL
Columbia University(Med)
Cook County, IL
Duke Hospital, Durham, NC
Elmhurst Hospital, Queens, NY
Emory University
Griffin Hospital, CT

Good Samaritan Hospital, IL
Harbor Hospital, Baltimore, MD
Hennepin County Minneapolis, MN
Jackson Park Hospital, Chicago, IL
Maricopa Medical Center
Meridia Huron Hospital, East Cleveland, OH
Mount Sinai Medical Center, FL
Providence Hospital, Washington, DC
Temple University, Conemaugh, PA
University of California, San Francisco, CA
University of Miami, Miami, FL
University of Texas,Houston Medical School, TX
VA Medical Ctr., Washington, DC
Wayne State University, Detroit, MI
West Virginia University Hospital, WV
Temple University Hospital