Active transport of intracellular cargo on cytoskeletal polymers via ATP‐driven motor proteins plays a key role in establishing well‐defined spatial patterns of functional intracellular components, which can range from proteins to big organelles like mitochondria. It is the interplay between active transport, diffusive movement in the cytosol and the geometry of the cell and its cytoskeleton that finally determines the distribution of the transported objects. To analyze such phenomena we extend our Green’s Function Reaction Dynamics (GFRD) framework to allow for an exact event‐driven simulation of active transport on microtubules and interactions with the cell membrane.