[Giggleswick (OP)]

Assuming a tube ran from one side of the earth to the other, through the center (setting aside all the logistical problems), and you fell in one side, how far would you fall? I assume gravity would stop you somewhere near the center of the earth. Would it matter if the tube ran along the earth's rotational axis versus perpendicular to it?

[syhprum]

If the tube ran thru the rotational axis of the Earth you would fall right thru up to ground level at the other side before falling back.
If the tube ran equator to equator you would try to do much the same but have trouble bumping the sides. 

[Phractality]

If you want to ignore all the practical matters, like temperature and air pressure, you should not choose planet Earth. Instead, talk about a hole thru the rotational axis of the moon. In that case, yes, you could fall in one side and out the other (assuming you don't bump the sides). As you near the center, your acceleration approaches zero. If the density is uniform, gravity is directly proportional to distance from the center and always toward the center.

Attempting the same stunt on Earth, pretending that you could have a hole thru the center, your main obstacles would be air pressure and temperature. The deaper you go, the greater the temperature and pressure and the lower the gravity. So you would quickly reach terminal velocity, perhaps 200 km/h; then your terminal velocity would get slower and slower. If you don't burn up first, you will fall perhaps 100 to 500 km before your terminal velocity is 10 km/hr; so that would last a few hours. (Just a wild guess; they don't make atmospheric temperature, pressure and terminal velocity charts for inside the Earth. The formulas for those values above the Earth are only approximate.) Even if you are insulated from the molten rock, the temperature of the air will incinerate you. Even if you're in an airconditioned capsule, you'll never reach the center of Earth.

[JMLCarter]

I think it could help to start at at least 1000ft up.

...and evacuate the hole.

[Giggleswick (OP)]

Ignore the pressure and temperature issues. I'm really more interested in the effects of gravity.