In the absolute gravity instruments developed by the Joint Institute for Laboratory Astrophysics (JILA) (Zumberge et al., 1982; Niebauer et al., 1986), the release of the dropped object induces systematic vibrations in the floor-gravimeter system. These vibrations can cause significant errors in the observed time-distance values from which the gravitational acceleration is computed. Detailed study of the vibrations affecting the gravity observations shows them to contain random noise and site dependent systematic components, which can be modeled by decaying sinusoids in the range of 10 to 120 Hz. This paper discusses (1) a mathematical filtering method to correct the observed time-distance array by identifying and removing all non-random signals from each individual drop, (2) upgrades of the gravimeter controller, which allow the collection of the data required to implement the mathematical filtering, and (3) mechanical filtering experiments using shock dampening pads and braces to minimize the vibrations. The maximum correction to observed gravity has been 23 μGal using the mathematical filter, typical corrections are in the 2-7 μGal range. The use of the shock dampening devices alone resulted in a three-fold reduction in the amplitudes and decay times of the systematic vibrations.