We have conducted a kinematic study of 165 young M dwarfs with ages of <~300Myr. Our sample is composed of stars and brown dwarfs with spectral types ranging from K7 to L0, detected by ROSAT and with photometric distances of <~25pc assuming that the stars are single and on the main sequence. In order to find stars kinematically linked to known young moving groups (YMGs), we measured radial velocities for the complete sample with Keck and CFHT optical spectroscopy and trigonometric parallaxes for 75 of the M dwarfs with the CAPSCam instrument on the du Pont 2.5m Telescope. Due to their youthful overluminosity and unresolved binarity, the original photometric distances for our sample underestimated the distances by 70% on average, excluding two extremely young (<~3Myr) objects found to have distances beyond a few hundred parsecs. We searched for kinematic matches to 14 reported YMGs and identified 10 new members of the AB Dor YMG and 2 of the Ursa Majoris group. Additional possible candidates include six Castor, four Ursa Majoris, two AB Dor members, and one member each of the Her-Lyr and {beta} Pic groups. Our sample also contains 27 young low-mass stars and 4 brown dwarfs with ages <~150Myr that are not associated with any known YMG. We identified an additional 15 stars that are kinematic matches to one of the YMGs, but the ages from spectroscopic diagnostics and/or the positions on the sky do not match. These warn against grouping stars together based only on kinematics and that a confluence of evidence is required to claim that a group of stars originated from the same star-forming event.