It depends a lot on the problem, but I tend to visualize the /processes/ of the programs that I am working on, rather than the code itself: for example, if I'm writing a function that parses a string, I try to imagine it going over the string, finding the positions of individual characters and capturing substrings between them. If I'm working with asynchronous or network code, I try to visualize how the processes (or hosts) should be working at different times, and how they might send messages to each other.
I find it very necessary and useful a lot of the time to draw diagrams, or even just written notes on the problem I am dealing with. A lot of my mental visualizations are very clear, but it is difficult to retain several at a time.
And like someone else in this thread shared, I often have to get away from the problem and do something else for my mind to garbage-collect itself and have enough room to clearly think up new angles. For me only a matter of even a few minutes away from the computer helps a lot.