Go to an engineering school and ask if calculus is required to be a good engineer ;-) You can be a programmer without a lot of math skills. You can do a good job in a number of different areas. You can be promoted and lead other programmers. You can write a lot of good code.
However, there are a lot of areas where knowing linear algebra will help you enormously. Many people who don't know linear algebra often don't see the problem because they will never choose a solution that will require it. They often don't know enough about it to realise that a solution exists and is potentially better than the solution they are reaching for.
Similarly, I can't tell you the number of times I've seen abject failures because the people involved did not understand statistics. In fact, if you only choose one math related area to learn about as a programmer, I highly recommend choosing statistics (which will unfortunately require calculus to understand well). Again, people who do not understand statistics often fail without realising that they are failing -- because they don't understand the statistics ;-)
I can make a similar remark for combinatorics and a variety of other mathematical disciplines. For a very cool job I once had to map animations onto a non-euclidean surface. Sure, I don't have to think much about math in my every day work wrestling with a legacy Rails system but I'm not sure I would want to define my entire career as doing that.
I would recommend that any programmer who wants to be a good programmer and to work in a variety of interesting fields to study math. Universities hardly have a monopoly on math. There are many good books and many internet resources to help you. You don't have to do it, but it will help you if you do.