We have obtained radial velocities of a sample of 18 ultracool dwarfs (M6.5-T8) using high-resolution, near-infrared spectra obtained with NIRSPEC and the Keck II telescope. We have confirmed that the radial velocity of Gl 570D is coincident with that of the K-type primary star Gl 570A, thus providing additional support for their true companionship. The presence of planetary-mass companions around 2MASS J05591914-1404488 (T4.5 V) has been analyzed using five NIRSPEC radial velocity measurements obtained over a period of 4.37 yr. We have computed UVW space motions for a total of 21 L and T dwarfs within 20 pc of the Sun. This population shows UVW velocities that nicely overlap the typical kinematics of solar to M-type stars within the same spatial volume. However, the mean Galactic and tangential velocities of the L and T dwarfs appear to be smaller than those of G to M stars. A significant fraction (~40%) of the ultracool dwarfs lie near the Hyades moving group (0.4-2 Gyr), which contrasts with the 10%-12% found for earlier type stellar neighbors. In addition, the distributions of the UVW components and the distributions of the total Galactic and tangential velocities derived for the L and T sample are narrower than those measured for nearby G-, K-, and M-type stars, but similar to the dispersions obtained for F stars. This suggests that, in the solar neighborhood, the population of L- and T-type ultracool dwarfs (including brown dwarfs) is kinematically younger than solar-type to early M stars, with likely ages in the interval 0.5-4 Gyr.