We present a new method for describing the electronic structure of graphene, by treating the honeycomb lattice as an arrangement of criss-crossing one-dimensional quantum wires. The electrons travel as free particles along the wires and interfere at the three-way junctions formed by the carbon atoms. The approach produces the linearly dispersive Dirac band structure as well as the chiral pseudo-spin wave functions. When vacancies are incorporated, the model reproduces the well known zero mode states.