Starting from an exact diagrammatic theory for density correlations in dense, atomic fluids, we derive a set of graphical approximations to this theory that are consistent with a set of physical assumptions that define an overdamped limit of the dynamics of the system. The results of a simple one loop approximation to this theory are then compared with data from molecular dynamics simulations for a number of correlation functions of a simple Lennard-Jones fluid at a single, high density and a range of temperatures. For correlation functions that have most of their decay over times for which the overdamped theory is valid, the one loop approximation gives accurate results, except for coherent correlation functions at small wavevector, for which the overdamped theory is not expected to be accurate. Although the temperature range we studied included only temperatures at or above the liquid's triple point, it is our hope that the overdamped theory can ultimately be used to characterize the dynamics of supercooled liquids. This will certainly require going beyond the one loop approximation.