Charge carriers of graphene show neutrino-like linear energy dispersions as well as chiral behavior near the Dirac point. Here we report highly unusual and unexpected behaviors of these carriers in applied external periodic potentials, i.e., in graphene superlattices. The group velocity renormalizes highly anisotropically even to a degree that it is not changed at all for states with wavevector in one direction but is reduced to zero in another, implying the possibility that one can make nanoscale electronic circuits out of graphene not by cutting it but by drawing on it in a non-destructive way. Also, the type of charge carrier species (e.g. electron, hole or open orbit) and their density of states vary drastically with the Fermi energy, enabling one to tune the Fermi surface-dominant properties significantly with gate voltage. These results address the fundamental question of how chiral massless Dirac fermions propagate in periodic potentials and point to a new possible path for nanoscale electronics.