Having BS/MS in Biomedical Engineering and finding my way out after a lot of struggle, I too went through this mind-boggling experience and was bombarded with advises on how screwed up I am but no real solutions. As most of the people here fail to realize that you are seeking a solution not some depressing crap, I am going to list a few real options:
1) Consider Clinical Engineering - It's a sub-branch of BME and there are a lot of opportunities. UConn offers MS in BME with concentration in Clinical Engineering which is a 2-year program with full-scholarship, paid stipend, and real-world experience at a participating hospital. When you graduate, you have a Master’s Degree, real experience, some bank balance and no freaking debt.
2) Consider being a technician just to get your foot in the door - There are many start-up to mid-size companies that repair and/or provide on-site services for medical equipment. Buying OEM parts could be costly and therefore healthcare facilities with large number of medical devices seek such options. All you need to know is basic electrical components, their behavior and some insight in technological background of a particular instrument. For example, to fix an infusion pump, you should know what's packed inside, how fluid flows, whats sort of mechanism prevents free flow etc and then you can determine if you need a new sensor or air-in-line assembly or a logic board. Most of this can come from your undergrad courses.
3) Consider staffing companies - We always appreciate some "help" finding a job.
4) Internships or Volunteer work - Great ways to make connection and sneak in.
5) Training and Placement companies - PrepMD - costly but considerable.
That's all I could think of right now. I usually go through stuff and put them onto this website: Home - BMEGEEK - Got a Question about Biomedical Engineering?
I know it's an old post and you might have settled down now. In that case, I would love to hear back from you about your experience. Write me at email@example.com