I'm guessing every school has a slightly different curriculum but I have had to take a very broad range of classes, many of them combining different aspects of math, chemistry, biology, biochemistry, physiology and physics. We also do some work with circuits/electronics but I don't especially like that area so I try to steer clear. It has been a very interesting major for me as I am fascinated by how the human body really "works" and how the body can be changed/altered/tweaked (ie, engineered). I also enjoy the design process of engineering and tinkering. That said, I am actually pursuing medical school but feel that my engineering training will give me a much different perspective when compared to the typical biology applicants. The one complaint I, and other undergrads in BME, have is that the BME major doesn't feel like a very niche major; we feel like a "jacks of all trades, master of none" major when compared to other engineering disciplines (EE, ChemE, EMA, etc.).
At my university we have to pick a BME emphasis in one area within BME and take a specific number of classes to fulfill a depth requirement during undergrad. I chose biomaterials/tissue engineering but other tracts include medical instrumentation, medical imaging, biomechanics, and health care systems and medical informatics. All of these tracts have strong ties to other engineering disciplines but include a heavy biology/human aspect which is what interests me. Classes I took this fall (senior year) were tissue engineering, stem cell bioengineering, biochemical engineering, engineering design with polymers, biology in engineering seminar, independent lab research and a capstone design class. There is so much variability that you can mix into the major if you chose to. I would definitely recommend BME for anyone interested in biology and engineering.
This will probably be a lot of info to look (and a bit overwhelming if you don't have anyone to walk you through it) but here is our undergraduate course packet. Skip the flowcharts and look at the text on pgs 1-4 & 14-17 for how my school defines BME and what courses we offer. Hope this helps a bit.
http://www.engr.wisc.edu/bme/pdfs/BMEUndergraduateBrochure.pdf