Use this page to generate SPICE kernels for the bodies of interest, and then use SPICE routines to calculate whatever you like to your heart's content.
Aarrr. I don't be knowin why ye be reinventing th' wheel.
Alrighty ya scurvy bilge rat, here it be:
$$x=a\left(\cos\tau-e\right)$$ $$y=a\sqrt{1-e^2}\sin\tau$$ $$z=0$$
That be givin it t' ye in th' plane. Then ya be rotatin th' plane with that land lubber Euler's transform in $\Omega$, $i$, $\omega$ (longitude of ascending node, inclination, and argument of periapsis), by multiplyin his matrix by yer vector up there:
$$\left( \begin{array}{ccc} \cos\omega \cos\Omega-\cos i \sin\omega \sin\Omega & -\cos\Omega \sin\omega -\cos i \cos\omega \sin\Omega & \sin i \sin\Omega \\ \cos i \cos\Omega \sin\omega+\cos\omega\sin\Omega & \cos i \cos\omega\cos \Omega-\sin\omega\sin\Omega & -\cos\Omega \sin i \\ \sin i \sin\omega & \cos\omega \sin i & \cos i \\ \end{array} \right)$$
Ye also be needin th' time it be ($\mu$ being th' $GM$ of the Sun):
$$t=\sqrt{a^3\over\mu}\left(\tau-e\sin\tau\right)$$
Yer $\tau=0$ and $t=0$ being yer time a periapsis. Ye be seein that ev'ry $2\pi$ in $\tau$, it be one a yer spins about th' Sun.
If yer scurvy rock be gettin close to a planet, then ye be wastin yer time, as th' orbit be changin on ye.