To start with something of an anti-answer, when I took courses in algebra (not linear algebra - I mean groups, rings, modules etc.) or representation theory as an undergraduate, I found it practically impossible to get anywhere by trying to visualize what was going on. I suppose, concurrent with other answers, the important thing is that I understood the material some other way.
By contrast, and as you say, some parts of combinatorics are certainly very visual. And (obviously?) topology (not so much first-course point-set stuff, but the real stuff) and differential (at least) geometry can both be very visual subjects. It can be a lot of fun trying to find ways to use geometric inituition to attack something that is ostensibly out of reach visually (e.g. in 4 dimensions or something not embedded in $R^3$ etc.)
At the moment I'm interested in geometric analysis, where I have come across some of the most pleasingly visual things yet.