Understanding the fundamental properties of graphene when its topography is patterned by the use of a compliant substrate is essential to improve the performances of graphene sensors. Here we suspend a graphene monolayer on SiO2 nanopillar arrays to form a puckered graphene-on-lattice and investigate the strain and electrical transport at the nanoscale. Despite a nonuniform strain in the graphene-on-lattice, the resistivity is governed by thermally activated transport and not the strain. We show that the high thermal activation energy results from a low charge carrier density and a periodic change of the chemical potential induced by the interaction of the graphene monolayer with the nanopillars, making the use of graphene-on-lattice attractive to further increase the electrical response of graphene sensors.