I've thought of that too. I think you would be in a perpetual fall. As you passed the core, your rate of fall would begin to decrease as you would then be "falling up". You should come to a complete and brief stop just under the surface on the other side of the earth and then you'd begin to fall down again, reversing the process endlessly.
Isn't that ignoring air resistance in the tunnel? I would think it would slow you down so that some of the energy caused by the first fall in would have been dissipated by friction, so you'd be like a pendulum that had no mechanism to keep it going, and each swing through the center would be shorter and shorter, till you stopped in the middle.
Of course, you'd burn up there if they didn't figure out the heat problem. Or, on your first plunge into the tunnel, you'd crash into the others whose pendulum had stopped. Science fiction speculation sure is fun, isn't it? I used to have all kinds of "intellectual discussions" with it back in high school, when my geeky buddies and I got together outside of electronics class!
After donning your special suit to allow you to withstand the temperatures, pressure, etc., you would gradually slow down as you neared the center of the earth and then stop completely and "float" in the core as the mass of the earth/gravity would be pulling you somewhat equally from all directions.
Sam, let's suppose you had a redneck-buddy on the other side who would, on the count of three and a toss of the beer can leap inside simultaneously with your ground-breaking experiment, what then?