A new, accurate, and nondiffusive method for three-dimensional advection of trace species is presented. The method preserves tracer structures by conserving the second-order moments of the spatial distribution of tracer during advection. Tracer concentrations are represented by separate, second-order polynomials within each grid box. This second-order moments method has been shown, in typical tests for advective schemes, to be equal or superior to presently available methods in terms of absolute accuracy, numerical diffusion, and computational cost.