Traditional methods for streamline tracing are mainly based on the Pollock Cartesian scheme and its extensions, which require dual grids to treat general tensor fields on unstructured meshes. The complexity and inflexibility of dual grids impose substantial limitations on the application of streamlines. In this paper, we propose a computationally efficient method to construct streamlines on the original grid, taking flow fields from arbitrary schemes including Galerkin finite element methods. An essential component of the proposed streamline construction is fast processing of non-conservative velocity fields to recover or maintain local mass conservation and normal flux continuity simultaneously. Our construction involves balancing conservation residuals on each pair of adjacent elements using a Gauss-Seidel type iteration. Locally conservative velocities are extended from element faces to interiors using a local single element mixed finite element method. Streamline and travel time are then obtained using standard methods of integration based on the Pollock algorithm. The proposed approach is shown to possess several advantages: it treats general tensor fields on unstructured and even nonmatching grids, maintains the optimal order of accuracy for high order elements, and avoids the difficulties imposed by nonphysical sinks and sources due to numerical inaccuracy. Several computational examples on structured and unstructured meshes are presented to demonstrate the effectiveness of the proposed method. Copyright 2005, Society of Petroleum Engineers.