How can time travel work by bending the space-time continum into a circle and then running around inside it?

I don't see how that would work. If space/time were bent back on itself wouldn't it be a state of stasis? What about causality?

There are certain geometries that allow for ``closed time-like curves''. Essentially, you can find paths in some metrics that would allow you to start at point A, travel a finite distance, and end up at the exact time and place that you started. You could see yourself leaving.

This is kind of what you said, but not really. I think there are other (bad) consequences of having a periodic time.

What bad consequences?

Kip Thorne's Black Holes and Time Warps is the definitive popular text on closed timelike curves. Read it. It's great!

I also found a very good New York Times article online: Remembrance of things future (http://www.nytimes.com/2005/06/28/science/28time.html?ex=1187841600&en=52d0c77d49426c6e&ei=5070), which includes this wonderful example (also discussed in Kip Thorne's book):

http://graphics8.nytimes.com/images/2005/06/27/science/0628-sci-4d-sub-WORMjp.jpg