"I just don't see how the heat content of the rock could affect the linear kinetic energy of the rock."

I didn't say it would.

As long as we are not taking things too seriously, I would offer the following reason why the initial rock would not swing up and down for very long.

I assume the earth's centre of gravity lies in the centre of the tube and that the rock will turn into viscous liquid due to heat as it passes through the earth. Call it the blob.

However those parts of the blob off the centre line will experience a horizontal component of gravity due to being off centre. This is clearer if you consider the instant when the blob is in the centre of the earth. A piece of the blob that is off centre will experience a slightly stronger gravitational attraction toward the side of the tube nearest to it than it will toward the side furthest from it. Due to this horizontal component the blob will gradually become more disc shaped than spherical. This trend will continue until the disc comes into contact with the tube wall. Once this starts happening things will quite quickly result in a ring of molten rock in contact with the tube in the centre of the earth.

 

SMF - I did say not really sure myself.