A computationally efficient, yet relatively simple Eulerian-Lagrangian method is proposed for solving the one-dimensional convection-dispersion solute transport equation assuming a steady or transient velocity field. The method uses a modified single-step reverse particle tracking (MSRPT) technique to handle steep concentration fronts. The scheme utilizes two weighting factors to control the movement of particles during a backward tracking step. One weighting factor greater than unity is used in the upstream region of the convection front, while another weighting factor less than unity is taken in the downstream region. The two factors were related empirically to the grid Peclet and Courant numbers. The MSRPT technique is carried out only within the concentration plume at each time step. For transient flow fields, the weighting factors were determined using an automatically adjustable procedure based on mass balance errors. The MSRPT method maintains the advantages of the traditional single-step reverse particle tracking (SRPT) procedure, i.e., producing efficient and oscillation-free calculations, but circumvents numerical dispersion introduced by SRPT. A large number of tests against analytical solutions for one-dimensional transport in uniform flow fields indicate that the proposed method can handle the entire range of Peclet numbers from zero to infinity. Numerical tests also show that the MSRPT method is a relatively accurate, efficient and mass-conservative algorithm for solute transport in transient flow fields. The Courant number at present cannot exceed 1. The MSRPT approach was found especially useful for convection-dominated problems; in fact, an exact numerical solution may be obtained with MSRPT for pure convection.