For 89 galaxies, mostly spirals, in the Virgo Cluster region, we have obtained optical long-slit major-axis spectra of the ionized gas. We find the following: (1) One-half of the Virgo galaxies we observed have regular rotation patterns, while the other half exhibit kinematic disturbances ranging from mild to major. Velocity complexities are generally consistent with those resulting from tidal encounters or accretion events. Since kinematic disturbances are expected to fade within ~10⁹ yr, many Virgo galaxies have experienced several significant kinematic disturbances during their lifetimes. (2) There is no strong correlation of rotation curve complexity with Hubble type, with galaxy luminosity, with local galaxy density, or with H I deficiency. (3) A few Virgo galaxies have ionized gas of limited extent, with velocities exceptionally low for their luminosities. In these galaxies the gas must be not rotationally supported. (4) There is a remarkable difference in the distribution of galaxy systemic velocity for galaxies with regular rotation curves and galaxies with disturbed rotation curves. Galaxies with regular rotation patterns show a flat distribution with velocities ranging from V₀ = -300 km s^-1 to V₀ = +2500 km s^-1; galaxies with disturbed kinematics have a Gaussian distribution that peaks at V₀ = +1172 ± 100 km s^-1, close to the cluster mean velocity. This latter distribution is virtually identical to the distribution of systemic velocity for elliptical galaxies in Virgo. However, disturbed galaxies are less concentrated to the cluster core than are the ellipticals; those near the periphery have velocities closer to the mean cluster velocity. Thus, spirals with disturbed kinematics are preferentially on radial orbits, which bring them to the denser core, where tidal interactions are strong and/or more common. Because they spend much time near apocenter, we observe them near the cluster periphery. Some may be falling into the core for the first time. These observations suggest that for a nonvirialized cluster like Virgo, galaxies may encounter either local (nearby galaxies) or global (cluster-related) interactions. These interactions may alter the morphology of the galaxy and may also play a role in driving the Virgo Cluster toward dynamical equilibrium.