Okay frist, all you dingbats who say "as you approach the center you would gradually slow down" I ask you this question: What experience in your life leads you to believe that as you crash toward the earth you slow down? Try jumping off a building and see if you gradually slow down. No. In fact, as you approach the center of the earth, the force of gravity acting on you would INCREASE and therefore you'd speed up. The radius (r) of the earth is about 6,378km. This means after you fell 3,200km, the force of gravity would be FOUR TIMES as strong as on the surface. Of course, in the center, you would appear "weightless" since all your parts would be pulled toward the center (silly as you'd most likely be squashed.
So, let's for the sake of argument say that you're squash-proof and there is no air in this heat-proof tunnel and therefore no air resistance. So we're tucked away in our safe pod poised on the lip of our pure-vaccum tunnel about to plummet toward the center of the earth. We tip forward and ZOOM!
Immediately we begin to plummet towards the center of the earth at 9.8 m/s^2 (G). Of course, you're in freefall so you feel weightless the entire time, even though your accelerating towards the earth at 1 G. This is similar to being in an elevator with the cables cut. If you had the presence of mind to have a scale with you and you weren't panicked about being flattened when you hit the basement, you could step on the scale and it would read 0.
After about 19 seconds of this pleasant weightlesses, you're travelling at about 11,000 m/s (or about 25,000 mph). This is when you ZOOM through the center of the earth. You are not gently slowling down here, you are going nearly 36 times the speed of sound (of course since it's a vaccum there is no sound).
Instantly the force of gravity switches to the reverse of your direction of travel and starts to PULL YOU BACK at 9.8 m/s^2 so that you are slowing down at exactly the same rate you accelerated at over the eact same distance (the radius of the earth). This time, however, you are travelling in the opposite direction of the force of gravity so you can sit down, read a book and relax for the next 19 or so minutes until you reach the surface of the earth.
When you get to the surface again, for the briefest moment your pod will stop and you may want to step out quickly because it's going to start hurtling toward the center of the earth again. And, yes, it will continue to do this for eternity. With no outward force on the pod and the force of gravity contstanly pulling on it, it will oscillate back and forth for the rest of time (or until the earth collapses). One round trip will take about 80 minutes and a one way will be about 40. You'll be weightless on your way "in" to the earth and a comfortable 1 G (your weight) on your way "out" to the surface.
Hope this clears it up for you.
* Note: What I didn't account for here, because I didn't have the time or the patience, was that G actually increases as you approach the center of the earth (it would be 4G at 1/2 the distance) I simply assumed a constant G. This means that your round trip times will be a little less and your final speed as you cross the center will be a bit more.
G = 9.8 m.s^2
r (earth) = 6,378.1 m