Summary: A new scheme is proposed for tracking the position of particles within a fixed mesh used to compute the motion of a continuous carrier fluid. The principal output of the scheme is the identification of the grid cell within which the particle resides at any time during the tracking of its trajectory. This information is required for the solution of the particle-motion equation and for modelling the exchange processes between the particle and surrounding fluid. The new method is explained in detail, and its performance is contrasted, by way of two examples, against a recently proposed highly efficient scheme. It is shown that the new method is about 30% faster than that scheme, held to be the most efficient algorithm documented in the literature.