We investigated changes to the linear viscoelastic properties of a mixture comprising polycarbonate (PC) containing 3 wt.% of a multiwalled carbon nanotube (MWCNT) and high-density polyethylene (HDPE), also containing 3 wt.% MWCNT, during post-processing annealing. The oscillatory shear moduli-i.e., the storage modulus G′ and the loss modulus G′′-gradually increased with residence time in a rheometer (i.e., annealing). The samples were prepared by compression-molding at low temperature, during which the MWCNTs became oriented by the applied squeeze flow. The marked increase in the oscillatory moduli may be attributed to the formation of a conductive MWCNT network owing to Brownian motion. Furthermore, the moduli increased more rapidly during high-temperature annealing. These increases can be expressed by a simple equation using only one characteristic time, i.e., the time required for MWCNT redistribution by Brownian motion. This characteristic time is considered one of the factors that control the structure of composites containing MWCNTs.