The properties of electronic transport in graphene superlattices which consist of symmetrical period potentials have been studied. It is found that such structures possess an unusual tunneling state inside the original forbidden gaps. Furthermore, the position of tunneling mode is highly dependent on the lattice constant and potential voltage but insensitive to period number, and a theoretical explanation for this phenomenon is proposed. Finally it is revealed that the electronic conductance is greatly enhanced and the Fano factor is strongly suppressed near the energy of the tunneling state. The characteristics of electron transport over symmetrical graphene superlattices may facilitate the development of many graphene-based electronics.