Large amplitude longitudinal (LAL) oscillations, consisting of periodic motions of prominence material along a filament axis, are rare but quite dramatic. The oscillations appear to be triggered by an energetic event, such as a microflare, subflare, or small C-class flare, close to a filament. Observations reveal speeds of several tens to 100 km/s, periods of order 1 hr, damping in a few periods, and displacements that are a significant fraction of the prominence length. We have developed the first self-consistent model for these oscillations that explains the restoring force and damping mechanism. We investigated the oscillations of multiple threads in our recent simulation (Luna et al. 2012), in which they form in long, dipped flux tubes through the thermal nonequilibrium process. The oscillation properties predicted by our simulations agree with the observed LAL behavior. In addition, our analytic model for the oscillations demonstrates that the restoring force is the projected gravity in the tube. Although the period is independent of the tube length and the constantly growing mass, the motions are strongly damped by the steady accretion of mass onto the threads. These suggest that a nearby impulsive event drives the existing prominence threads along their supporting tubes, away from the heating deposition site, without destroying them. As is also the case for newly formed condensations, the subsequent oscillations occur because the displaced threads reside in magnetic concavities with large radii of curvature. Our model yields a powerful seismological method for constraining the coronal magnetic field and radius of curvature of dips. Furthermore, these results indicate that the magnetic structure is most consistent with the sheared-arcade model for filament channels. We conclude that the LAL movements represent a collective oscillation of a large number of cool, dense threads moving along dipped flux tubes, triggered by a small, nearby energetic event.