The so-called artificial graphene is an artificial material whose low-energy carriers are described by the massless Dirac equation. Applying a periodic potential with triangular symmetry to a two-dimensional electron gas is one approach to make such a material. According to recent experimental results, it is now possible to realize artificial graphene in the lab and to even apply an additional lateral, one-dimensional periodic potential to it. We name the latter system an artificial graphene superlattice in order to distinguish it from a genuine graphene superlattice made from graphene. In this study, we investigate the electronic structure of artificial graphene superlattices, which exhibit the emergence of energy band gaps, merging and splitting of the Dirac points, etc. Then, from a similar investigation on genuine graphene superlattices, we show that many of these features originate from the coupling between Dirac fermions residing in two different valleys—the intervalley coupling. Furthermore, contrary to previous studies, we find that the effects of intervalley coupling on the electronic structure cannot be ignored, irrespective of the length of the spatial period of the superlattice.