A new algorithm for the calculation of particle trajectories is introduced. The algorithm combines second-order accuracy in the real frequency with third-order user-adjustable attenuation. It requires little storage of data from previous time levels. The method was designed for use in implicit particle-in-cell plasma simulation codes, and this application is treated in detail. It may also prove useful in other applications where one seeks to preserve the accuracy of low-frequency oscillations while rapidly damping under-resolved high frequency motions, e.g., solution of the field equations in electromagnetic particle codes. An explicit variation, wherein future quantities are obtained by extrapolation, can provide attenuation but not large-timestep stability.