How do we determine the orbital period of an extrasolar planet?

Answer 1

Via doppler shifts in the parent star's spectrum. These indicate the velocity of the star relative to the Earth. If the velocity oscillates towards and away from us over time, it is orbiting some center of mass external to the star's center of mass.

In other words, whatever is orbiting the star, makes the star move back and forth a bit in response. That is the ol' "for every action there is an opposite and equal reaction" rule kicking in.

This guy 'splains it in his class notes at a layman's level:
http://www.astro.washington.edu/larson/Astro150b/Lectures/ExtraSolarPlanets/xsp.html
Here is a catalog of the planets found using the doppler technique (there are a couple of others, but they are not as fruitful. . . yet)
http://exoplanet.eu/catalog-RV.php

You can click through the catalogue to find what the data looks like, but here is one bone, since it can be a bit difficult if you are not familiar with astronomy and exoplanets
http://exoplanets.org/esp/55cnc/55cnc.shtml

Answer 2

look for a string hanging from the bottom of the planet and count 28 days from that point