I did an engineering degree (not comp sci), then after I graduated I just read a few programming books and got a job writing software. I worked for a couple of years before going back to university to get a graduate diploma in computer science (basically half a masters).
Although I already basically knew everything that was taught in the comp sci diploma after working for a couple of years, having all the basic concepts laid out formally instead of me just googling bits and pieces (well, altavista-ing back then) really helped a lot. I would say it is definitely worth doing the extra courses as background info, as Scott said.
A lot of the modern languages you might end up working with hide the low level detail, but even so it makes a huge difference to your understanding of what is going on when you know about logic gates and flipping bits and stack vs heap memory and things like that.
No matter how much you learn though, you should still be prepared to learn just about everything you need to know on the job anyway. It always works out that way.
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