I have recently started using Twitter for promotions.
Let's say you can break your customers into three specific niches: A, B, and C.
While I perform the same general service for each, the way I market to each group is entirely different. So if I had one twitter account called "Website" then I doubt the A's would want to hear tweets about stuff the C's would find interesting.
So is the solution to have three separate Twitter accounts? Website_A, Website_B, and Website_C? This solves the problem of providing them with targeted tweets, but raises a new problem of having to gain followers for three accounts versus one.
I think your assessment that you need three accounts is correct due to lack of cross-interest. If you promote each one in the venues where you interact with the different groups, you'll get natural growth. You could also use a tools-based approach (e.g., keyword-based following) if you want to get aggressive. That assumes you can segment the groups in some way.