Jon Shock writes at SciTalks about visualizing things in four dimensions, although it's actually less about visualization than about how we don't need to visualize so long as we can calculate -- although he does provide a link to a youtube video showing how one can obtain the projection of a hypercube onto a plane.
I've never been good at visualizing things in more than three dimensions (actually, to tell the truth, I have trouble visualizing things in my head in more than two dimensions; to visualize anything successfully in three dimensions I have to use my hands and point at things in the empty space in front of me). But a guy I knew in college claimed to be able to visualize things in five dimensions fairly easily. The fourth dimension was time -- so a four-dimensional sphere, for example, can be thought of as a three-dimensional sphere which grows from a point to a large ball and then shrinks back again. The fifth dimension was color, although I never really understood this (he wasn't very good at explaining it); I think the idea was that, say, red would correspond to a large coordinate in the "color" dimension and blue would correspond to a small coordinate, similar to weather maps showing temperature. So like weather maps allow us to visualize a surface in three-dimensional space with a two-dimensional picture, you can get an extra dimension this way (or even more than one! -- the space of colors is usually thought of as three-dimensional -- although I can't imagine really taking advantage of that).
I suppose a lot of this visualization ability comes with practice, though, and I rarely have any reason to think in more than three dimensions. I can definitely see it coming with practice when I teach calculus students about solids of revolution; at first they just don't seem to get it but eventually they seem to figure it out. The difference there is that I can tell them to draw a picture of the solid in question, and they draw a two-dimensional picture of a three-dimensional thing, which is something we're all used to; drawing a two-dimensional picture of a four-dimensional thing that doesn't throw away all the useful information is a lot more difficult, to the point where perhaps you really have to understand what the thing looks like before you can draw the picture. So they're useful for explaining results to other people, but not so much for coming up with new results.