Linear and nonlinear viscoelastic measurements were utilized to probe transient morphological changes in a melt-blended polypropylene-clay nanocomposite containing 3 wt % organically modified montmorillonite clay, 10 wt % maleic anhydride functionalized polypropylene (1% maleic anhydride content), and 87 wt % polypropylene homopolymer. Steady shear rate sweeps show that the viscosity decreases monotonically with shear rate, characteristic of a mechanically percolated material with a yield stress. Subsequent sweeps following annealing of up to 2 h show that the viscosity at low shear rates remains substantially lower than the initial sweep, indicative of slow or arrested organoclay disorientation. In startup of shear flow, as-processed and presheared nanocomposite samples both exhibit increasing viscosity overshoots with annealing time for up to 6 h, although the overshoots for the presheared samples remain uniformly smaller than those of the as-processed material. Similarly, small-amplitude oscillatory shear experiments on as-processed and presheared samples also reveal that the storage modulus and complex viscosity increase logarithmically with time, while the loss tangent declines steadily. Both sets of data point to increasingly solidlike rheology over time, a phenomenon that is discussed in the context of soft glassy dynamics.