Remember this thought experiment?
Imagine a perfectly straight hole drilled through the Earth, passing directly through the center. Now imaging falling into this hole. What would happen?
The answer is - you would fall straight down, passing the center of the Earth, and emerge on the other side. As you fall toward the center the force of gravity diminishes, at the center the force is zero, and as you swing out the other side gravity increases. So you swing back and forth through the hole, and depending on air resistance and friction (hitting the sides of the hole :) you'd finally settle in the center.
Liron Shapira has written a cool little program which models gravity; if you download this program and run it, you can simulate this case pretty easily. Very cool.
(Liron, I hope you don't mind that I posted your program; it appears Angelfire no longer allows EXE downloads.)
[ Later: Liron emailed to provide this updated link. Thanks! ]
With this program you can also model another interesting thought experiment:
Imagine a train which goes on a perfectly straight track between two cities. Perfectly straight, as in, through a tunnel drilled into the earth (to compensate for the Earth's curvature between the two cities). What would happen?
The surprising answer is - the train would be powered by gravity down into the tunnel, reaching maximum speed halfway between the two cities, and would then slow down just as it reached the other side. Neglecting air resistance and friction, the train would not need any other source of power to travel between the two cities.
As far as I know, this phenomenon has never been exploited for actual travel.