There have been a ton of really accurate thighs here. As someone who fit this bill and had some difficulties, I'll point out two things that I would have done much differently.
1. Applying to almost all top schools. I went through the MSRE before applying and did the stats to figure out where I would be most likely to get in as an out of stater, because a vast majority of the schools accepted like 98% of their students from in state, and I didn't have a billion med schools like people from Massachusetts do. But the schools that really don't care where you come from are either at the very top of the list or the very bottom in terms of reputation, and of course I didn't really want to go to somewhere at the very bottom of the list if I could avoid it. I figured that with my application (which was quite well rounded, no glaring omissions, certainly) I was fine with doing 25 schools even with most of them being top schools. And I got interviews. But I'll tell you, when you interview at Harvard, everyone has those grades. And while I had some research and a pub, it's hard to compete with first-author Nature pubs at a school like that. And of course, when other schools see that you have a ton of top tier schools and then you're interviewing with them, they automatically assume it's as a fallback, so they either don't give an interview at all or waitlist you instead of accepting you, unless you specifically show a lot of interest in them. Of my 7 interviews, 4 were top 10, 2 were top 20, and one was top third. No one else bothered to interview me because they figured I wouldn't go there. In the end, I got rejects mostly and a couple waitlist (from the lower two on the list) and had to send a letter to the one I preferred to get that waitlist turned into an acceptance. Of course, maybe I'm a horrid interviewer, but everyone I practice interviewed with thought I did great, and these were people who do interviews for a living. (Well, there was a glaring issue with that looking back, which is my #2 point. But I honestly don't think it would have made much of a difference.) Seriously, diversify that school list or the numbers game will kill you no matter your stats.
2. Bad advice. Don't get interview advice from your PhD PI. Don't get it from doctors that aren't involved in admissions. Get it from recent acceptances to med school and faculty involved in interviewing. Goro mentioned above not to do "show and tell," bringing in a portfolio. Guess what my PI told me he loves to see when he sees candidates for grad school come in? Yup, a research portfolio, to show they're prepared and taking it seriously. And I stupidly thought it was a good idea, because all the other people I asked about it also had no experience interviewing med students. In the end, I think it felt gimmicky and/or that I was trying to make up for a deficit by bringing that in. Goro is spot on; it's in your application, at least for the most part, so don't do stuff like this. But while I do think it hurt me, I don't think it was an automatic excluder from any school. It's just that if you're going to go to schools where everyone is a stellar applicant, you really can't have ANYTHING make you seem odd. Even if you do seem perfect, again, the numbers game will get you.
Hope that helps!