If we are right at the earth's centre of mass (or centre of gravity), I believe nothing will happen, as the resultant force on you is zero.

This assumes I am a point-mass (a dimensionless point in space).

Imagine we split the earth in half. The force created by the mass of the earth pulls us down towards it, its centre of mass. However, if we add the other half, bearing in mind the assumptions, we can tell that the mass of the other half must also be pulling on us with the same magnitude, but in the complete opposite direction, thus creating a resultant force of zero. This method can be used in all 3 planes: all the forces result to zero, as there is always the same amount of mass on the other side, thus the same amount of force pulling in the opposite direction.

In reality however, we are not a point mass. This means that there are some moments involved, i.e. you start rotating. However, these moments would be incredibly small compared to the distances involved (the radius of the earth) thus the rotating force will be extremely small.

But why doesn't the earth pull you apart then? Or crush you? If one force is pulling you one way, and the other another, doesn't that mean you tear apart?

No. Gravity pulls on every single atom in your body. So therefore each and every single atom in your body has a resultant force of zero. If it were to tear you apart, gravity would have to only pull on specific atoms, such as one force pulling you arm one way, and the other force pulling your torso the other way.

I hope this helps. Please don't hesitate to point out any errors, or anything you don't understand.