Let's assume we really could dig a tunnel all the way through the Earth's core, right to the other side. If we dropped a rock at one end, the first thing that would happen would be an acceleration toward the centre. It would get pulled by the gravity of an entire planet and hurtle down at 9.81 ms-2. But here's where it gets interesting.
The deeper you got into the Earth, the more of the planet would be above you. As the rock falls, it gradually finds more and more of the Earth is now above it (rather than below it), and all of this plant-stuff has a gravitational attraction of its own. This means as it started to approach the centre of the Earth it would begin to slow down, it starts to get pulled "up" as well as "down".
It's going to have a lot of momentum however, which will carry it through the centre of the tunnel, but it's already begun slowing down, so it won't go shooting to the other side of the planet. What will happen is that it will begin to yo-yo around the centre of the Earth, being pulled back and forth for several minutes until eventually it reaches an equilibrium and will then sit in the centre, hovering perfectly. At this point, it is equally attracted outwards (because all the gravity is surrounding it) so it will become weightless and will sit there, floating, until we drop another rock down in an attempt to knock it out.