A variable explicit/implicit numerical method is presented for calculating fluid advection on hybrid unstructured meshes. The method is intended for unsteady flows that have a large range of advective time scales, with subregions where the flow may be quasi-steady on time scales of interest. The method generalizes the method of Collins, Collela, and Glaz by providing an explicit-to-implicit switching criterion that is suitable for use on unstructured meshes. No directional splitting is used; upwinding is in the direction of the material velocity, and thus we define and use a new multidimensional limited gradient that generalizes Van Leer's MUSCL limiting [7]. After reviewing the one-dimensional method of Collins, Collela, and Glaz, we present the generalization to hybrid unstructured grids and detail the results of four example calculations that demonstrate the accuracy and flexibility of the new method.