The cytoplasm is a highly complex and heterogeneous medium that is structured by the cytoskeleton. How local transport depends on the heterogeneous organization and dynamics of F-actin and microtubules is poorly understood. Here we use a novel delivery and functionalization strategy to utilize quantum dots (QDs) as probes for active and passive intracellular transport. Rapid imaging of non-functionalized QDs reveals two populations with a 100-fold difference in diffusion constant, with the faster fraction increasing upon actin depolymerization. When nanobody-functionalized QDs are targeted to different kinesin motor proteins, their trajectories do not display strong actin-induced transverse displacements, as suggested previously. Only kinesin-1 displays subtle directional fluctuations, because the subset of microtubules used by this motor undergoes prominent undulations. Using actin-targeting agents reveals that F-actin suppresses most microtubule shape remodelling, rather than promoting it. These results demonstrate how the spatial heterogeneity of the cytoskeleton imposes large variations in non-equilibrium intracellular dynamics.