We present the kinematics of 35 highly r-process-enhanced ([Eu/Fe]>=+0.7) metal-poor (-3.8<[Fe/H]< -1.4) field stars. We calculate six-dimensional positions and velocities, evaluate energies and integrals of motion, and compute orbits for each of these stars using parallaxes and proper motions from the second Gaia data release (Cat. I/345) and published radial velocities. All of these stars have halo kinematics. Most stars (66%) remain in the inner regions of the halo (<13 kpc), and many (51%) have orbits that pass within 2.6 kpc of the Galactic center. Several stars (20%) have orbits that extend beyond 20 kpc, including one with an orbital apocenter larger than the Milky Way virial radius. We apply three clustering methods to search for structure in phase space, and we identify eight groups. No abundances are considered in the clustering process, but the [Fe/H] dispersions of the groups are smaller than would be expected by random chance. The orbital properties, clustering in phase space and metallicity, and the lack of highly r-process-enhanced stars on disk-like orbits, indicate that such stars likely were accreted from disrupted satellites. Comparison with the galaxy luminosity-metallicity relation suggests M_V_>~-9 for most of the progenitor satellites, characteristic of ultra-faint or low-luminosity classical dwarf spheroidal galaxies. Environments with low rates of star formation and Fe production, rather than the nature of the r-process site, may be key to obtaining the [Eu/Fe] ratios found in highly r-process-enhanced stars.