The graphene grain boundaries (GGBs) of polycrystalline graphene grown by chemical vapor deposition (CVD) typically constitute a major reason of deterioration of the electrical properties of graphene-based devices. To reduce the density of GGB by increasing the grain size, CVD growth conditions with a reduced CH4 flow rate have been widely applied and, recently, electropolishing of copper (Cu) foil substrates to flatten the surface has been undertaken prior to graphene growth. In this study, we show that polycrystalline graphene layer grown on typical Cu foil features two heterogeneous regions with different average grain sizes: small-grain regions (SGRs) and large-grain regions (LGRs). Statistical analysis of the grains of the graphene layers grown under different process conditions showed that SGRs (which form on Cu striations) limit the average grain size, the ability to control the grain size through adjustment of growth conditions, and global grain-size uniformity. Analysis showed that the surface-flattening process significantly improves grain-size uniformity, and monolayer coverage, as well as the average grain size. These results suggest that a process for flattening the surfaces of Cu substrates is critical to controlling the quality and uniformity of CVD-grown graphene layers for practical device applications.