X-ray microtomography is routinely used to image the three-dimensional pore space of sedimentary rocks. Flow and transport properties can then be simulated directly in such images. Advective transport in porous media is frequently simulated using streamlines. We present a novel streamline tracing algorithm based on a substantial development of the most widely used method (the Pollock algorithm) employed for macroscale (Darcy) flow, making it consistent with solutions of the Navier-Stokes equation with no flow at solid boundaries. We use this new algorithm to calculate breakthrough curves and time-of-flight distributions for advection-dominated transport in two three-dimensional images of sedimentary rocks containing up to voxels: a sandstone and a carbonate. We show that our approach provides a more accurate description of flow, particularly when only a few image voxels span each pore. Therefore, it is better suited to capture anomalous (non-Fickian) transport behaviour than the standard Pollock method.