Here is a link to NIHs definition of their nanotech programs, which I am sure are woefully underfunded... but that aside.. I personally view nanotech as anything that is non-organic at the submicron (nano) size scale. In the context of Medical applications would be their applications to the human body in particular.
I work in Drug Delivery, so I know a little bit about this stuff. And what I was getting at was whether the OP meant non-organic...i.e. the electronic devices (biosensing, etc.) or more along the scaffold/colloidal particles stuff. Essentially, my point is that by saying nanotech you could get a wide variety of research experiences.
It's interesting that you mention Tissue Engineering, a lot of the approaches are chemical/organic (natural materials, biocompatible polymers, etc.), so again I'm not quite sure what you're getting at, as you seem to suggest the OP meant electronic devices stuff.
Anyway, BME's definitely a good choice no matter what aspect you want to get into, and you're right, its very interdisciplinary. Either way you're going to get a spectrum of approaches, it's just the emphasis will vary from lab to lab, field to field, and saying nanotech is very broad.
Also, let me just say IMHO its a big bonus to be an MD/PhD in this field, lots of engineers don't know their physiology/anatomy as well as they should and it helps keep your research more clinically relevant. Lots of good ideas, not enough translation, it seems. And you're close enough to the clinic to make the clinical expertise worthwhile.
To the OP, let me give you a list of schools that are known for dealing with this stuff. This is only my personal experience from my time in field (relatively limited), and its certainly not comprehensive and may be very biased/colored by my own experiences:
Tissue Engineering: Harvard/MIT (liver project, Jay Vacanti), Pittsburgh (big transplant center too)
Devices/Implants: Hopkins (though they're best known for computational/modelling stuff, I think), CalTech, Berkeley (lots of BioMEMs stuff at these last two), UCSD
Drug Delivery: MIT (Bob Langer), Penn, Yale, Duke, Georgia Tech