Although it is only based on my own experience, so take it with a grain of sale, I think a short and strong Letter of Intent sent to ONE of your waitlists can be powerful.
I think Letters of Interest scream: "I like your school, but not enough to commit." My gut tells me that those letters could actually be harmful at this stage because they assume you might be writing a "commit" letter to another school.
Content and timing are important for the Letter of Intent. And if you have been accepted by other schools, TELL them that in the letter: "I have been accepted by 4 schools and plant to matriculate to Med School X." That way you do not sound desperate and it rings true that you really want their school. (And if you are lucky enough to hold an acceptance to a higher ranked school, tell them you will withdraw from that if they accept you to their school. That really proves you love them for the right reasons -- not just ranking.)
Of course, everything you say has to be TRUE!!!
Oh, of course have people who might have influence call. Those people usually really want to be able to say: "Joe, take this kid. You are his first choice and he has told me he will matriculate to your school if accepted."
