The collisional relaxation of vibrationally highly excited CS2 by thermal ensembles of H-2, CO, HCl, CS2, and CH4 is investigated theoretically using classical trajectory calculations with empirical potentials that include van der Waals and electrostatic interactions. The successive collisions method suggested by Bruehl and Schatz is shown to be an effective way to study energy transfer from vibrationally hot polyatomic molecules to molecules of a thermal bath. The average energy transferred per collision to H-2 and CO matches experimental measurements within a factor of 2, while that to CH4 and CS2 matches experiment within a factor of 4. These differences are consistent with combined theoretical and experimental uncertainties. Energy transfer to HCl is, however, off by as much as a factor of 7, despite our use of a potential which is realistic at long range. The energy transferred is partitioned mostly to relative translation when the collider is H-2 or CO and partly to rotation for CH4. The energy dependence of the energy transfer and the interrelations among energy-transfer moments, including <DELTA-E(up)>, <DELTA-E(down)>, and <DELTA-E2>, are also reported.