We present Magellan/IMACS spectroscopy of three recently discovered ultra-faint Milky Way satellites, Grus II, Tucana IV, and Tucana V. We measure systemic velocities of vhel=-110.0 ± 0.5 km s-1, vhel= 15.9-1.7 +1.8 km s-1, and vhel=-36.2-2.2 +2.5 km s-1 for the three objects, respectively. Their large relative velocities demonstrate that the satellites are unrelated despite their close physical proximity. We determine a velocity dispersion for Tuc IV of σ = 4.3-1.0 +1.7 km s-1, but we cannot resolve the velocity dispersions of the other two systems. For Gru II, we place an upper limit (90% confidence) on the dispersion of σ < 1.9 km s-1, and for Tuc V, we do not obtain any useful limits. All three satellites have metallicities below [Fe/H] = -2.1, but none has a detectable metallicity spread. We determine proper motions for each satellite based on Gaia astrometry and compute their orbits around the Milky Way. Gru II is on a tightly bound orbit with a pericenter of 25-7 +6 kpc and orbital eccentricity of 0.45-0.05 +0.08. Tuc V likely has an apocenter beyond 100 kpc and could be approaching the Milky Way for the first time. The current orbit of Tuc IV is similar to that of Gru II, with a pericenter of 25-8 +11 kpc and an eccentricity of 0.36-0.06 +0.13. However, a backward integration of the position of Tuc IV demonstrates that it collided with the Large Magellanic Cloud at an impact parameter of 4 kpc ∼120 Myr ago, deflecting its trajectory and possibly altering its internal kinematics. Based on their sizes, masses, and metallicities, we classify Gru II and Tuc IV as likely dwarf galaxies, but the nature of Tuc V remains uncertain.