Could you feel your weight falling through the a tube drilled through the center of the earth? Suppose you drill a hole through the center of the earth (assume the earth is uniform and no air resistance) and you jump in. Would you be "weightless" throughout the entire fall?
The reason I ask is that if you jump off a cliff, throughout the entire fall you feel weightless (just like when astronauts train for the weightless feeling in orbit, they practice by going in an airplane and having the airplane fall to approximate the experience). Does this same weightless experience happen when you are falling through the center-of-the-earth tube?
I know that if you are stationary at the center of the earth, then you are weightless; but, I'm interested in falling throughout the entire hole.
The reason why I'm confused is that it's well-known that when you fall, you oscillate (simple harmonic motion) up and down the tube and this oscillation seems to imply that you will feel your weight.
 A: In order to feel your weight something must be present that prevents you from free falling.
In everyday life the ground we are standing on provides the barrier that keeps us from free falling.

You refer to the implication that an object in free fall inside a corridor straight through the center of gravity of a gravitating mass will undergo oscillation. That is: the direction of gravitational acceleration is in opposite direction on opposite sides of the center of gravitational acceleration.
The point is: if you grant that inertial mass and gravitational mass are equivalent then an accelerometer-strapped-to-the-object will not register change of direction of gravitational acceleration, since given inertial-gravitational mass equivalence the accelerometer will not register any acceleration in the first place!
If equivalence of inertial and gravitational mass is granted then you expect that an accelerometer-strapped-to-the-object will register acceleration only if some barrier is preventing free fall.
A: You would experience the feeling of weightlessness throughout the entire vibration as you move back and forth through the Earth.  You hint at this in your question when you make reference to astronauts training for weightlessness in aircraft.  The aircraft in this case is not simply moving down (and horizontally) but rather flying in a parabolic arc.  Throughout the entire up and down motion the passengers experience the same sensation of weightlessness.  You could think of this as half of an oscillation.
As explained in Cleonis's answer weightlessness will be experienced anytime the force of gravity is the only force acting on a person.
A: 
Suppose you drill a hole through the center of the earth (assume the earth is uniform and no air resistance) and you jump in. Would you be "weightless" throughout the entire fall?

Yes, because you would follow the spacetime geodesics. The case you described has been precise analyzed by Edward Parker in A relativistic gravity train, where he calculates the trajectory of a test particle under the gravitational influence of a ball with
uniform mass density as the particle falls through the ball’s diameter.
