Dear InfernoSun & rocketboy:
Personally, I consider your questions valid, but then I am not blessed with fixed focus. Fundamentally, DuncanM's question, "What gets you excited?", was very valid. To be truly successful in your career, and I'm measuring that from the way you will look back on it, you have to not only be willing, but be completely and fully committed to problem solving. As they say, you must be willing to eat, live, and sleep with problem solving.
It is true that most accountants work in the accounting departments, doing some aspect of accounting, very similar to the human resource types. My experience is probably far different than the others chiming in, for I was graduated with a Bachelor of Science in Physics from a major US university. Prior to that I had attended an Aircraft and Engine (A&E) Mechanic's school and worked as an aircraft electrician and technical writer. Upon graduation, I obtained a job with a major space engineering firm, not a bird farm, Hughes Aircraft Company. I worked on testing, mechanical and electronic, spacecraft instrumentation, and the first gyrostat. I took additional evening courses in engineering subjects, such as stress analysis, machine language programming, and researched in the library the latest progress in the fields of tribology, a subject with which engineers were totally ignorant. I became involved in ball bearing technology and the mathematics related thereto. Because I lived, breathed, and slept what I was working on, I was able to readily identify the source of problems when they occurred and mathematically prove my point if needed.
During my career, I have worked in electromechanical design and test, control system engineering, testing of various and asundry control system components, testing of completely integrated spacecraft (in and out of a thermo-vacuum chamber). At one point, I instigated the surreptitious integration of personal computers into a testing laboratory so as to automate the testing process. I was involved in the performance testing of the first Global Positioning System (GPS) receivers and ended up developing a large experience with GPS.
Due to my interests in performing Fourier Analysis on the noise produced by operating rolling element bearings, primarily ball, when Wavelets emerged, I dived into studying the subject. Admittedly, the mathematics did give me many problems, but I continued to pursue the subject.
Based upon my experience, to be anywhere successful, you must be fully committed, body and soul. Like any profession, you are at the whims of economic conditions, et al. Whatever you decide to do, you should try to obtain as much of a mathematics background as you can.
Unfortunately, life's crystal ball is too fractured to read, you will never really know what area of engineering that you will have to adapt to, but a strong mathematics background coupled with a broad understanding of fundamental physical principles will provide you with an ability to bring yourself up to speed, independent of the situation.
Best of Luck
JoeO