In spite of the very wide body of research carried out in the last decades, Polyolefin-based nanocomposite are still an important scientific as well as technological topic. In particular, polypropylene-inorganic nanocomposites have gained increased attention due to their improvement in properties such as high modulus, increased strength and heat resistance, decreased gas permeability and flammability. Among the different experimental techniques, rheology is an important tool to assess the properties of such systems. On the one hand, rheology provides important information on the nanocomposite processability. On the other hand, a more sophisticated use of the rheological techniques provides a very useful tool to explore the system nanostructure. In this work, we measure the linear viscoelastic and the non-linear shear flow rheology of some polyolefin-based commercial materials that have been filled with functionalized Layer Double Hydrotalcite (LDH) nanoparticles. The linear viscoelastic measurements show the classical signature of a nanoparticle network, indicating that a good degree of dispersion has been obtained. More interestingly, and unlike the pure polymer, the nanofilled materials show sustained stress oscillations during the start-up of shear flow, whose occurence is related to the flow history imposed to the material. This suggests that the orientation of the LDH nanoplatelets changes with time and is controlled by nanoparticle diffusion.