In this work, cocontinuous blends of linear low density polyethylene/ethyl vinyl acetate containing graphene (GN) have been studied. Although mass-produced GN grade prepared by mechanochemical exfoliation of graphite and a facile melt compounding technique were adopted, it was possible to lower the electrical percolation threshold significantly by controlling the localization of GN nanoplatelets in the blend and by applying an appropriate thermal annealing procedure. The electrical and rheological properties of the obtained nanocomposites were systematically investigated to get an insight into the composite morphology. During annealing, an alignment between time-dependent behaviors of viscoelastic moduli and electrical conductivity was observed. An increase of both quantities and a simultaneous coarsening of the blend's morphologies occurred during the first 30min of annealing followed by a more stable behavior. This rise was attributed to the diffusion and flocculation of GN nanoplatelets and their migration to the interface. Furthermore, the electrical and rheological percolation threshold concentrations were evaluated using a scaling power law. The electrical percolation threshold was reduced to 0.5vol.% upon thermal annealing and was close to the rheological percolation threshold. Finally, the viscoelastic response of the composites was well described by a two-phase model, indicating that the effect of the relaxation dynamics of the interfacial network does not depend on the blend's morphology, even though the latter affects the space arrangement of GN and consequently the strength of the formed network.