The cytoplasm is a highly complex and heterogeneous medium that is structured by the cytoskeleton. Cytoskeletal organization and dynamics are known to modulate cytoplasmic transport processes, but how local transport dynamics depends on the highly heterogeneous intracellular organization 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 transport dynamics in different sub-cellular environments. Rapid imaging of non-functionalized QDs revealed two populations with a 100-fold difference in diffusion constant. Depolymerization of actin increased the fast diffusing fraction, suggesting that slow QDs are trapped inside the actin network. When nanobody-functionalized QDs were targeted to different kinesin motor proteins and moved over microtubules, they did not experience strong actin-induced transverse displacements, as suggested previously. Only kinesin-1 bound QDs displayed subtle directional fluctuations, because the specific subset of stable microtubules used by this motor underwent prominent undulations. Using actin-targeting agents we found that the actin network suppresses most microtubule shape remodeling, rather than promoting them. These results demonstrate how the spatial and structural heterogeneity of the cytoskeleton imposes large variations in non-equilibrium intracellular dynamics.