Here's how you do it:

Force = -m M G r^{-2},

where m is your mass, M is the mass contained within a spherical shell under your feet, at any given depth, and r is how far you are from the center.

You can compute M=4/3πr^{3}ρ,

where ρ is the mass density of the earth (you assume it to be uniform, but whatever... this is a back-of-the-envelope computation).

ρ=M_{e}/(4/3π R^{3}),

where M_{e} is the earth's mass.

Plugging all this back in

F=-mM_{e}GR^{-3}r.

All that junk to the left of r is constant. This is identical to the force exerted by a spring, F=-kx, where k is given by all that constant stuff. The period of a spring is 2π(k/m)^{-1/2}, so the period of your oscillations through the center of the earth is

(M_{e}GR^{-3})^{1/2}

I get ~250 seconds transit time. I might have made a slight mistake, but your motion through the earth is just like a spring.