  We investigate the effect of a periodic potential on the electronic states
and conductance of graphene. It is demonstrated that for a cosine potential
$V(x)=V_0\cos(G_0x)$, new zero energy states emerge whenever $J_0(\frac
{2V_0}{\hbar v_F G_0})=0$. The phase of the wavefunctions of these states is
shown to be related to periodic solutions of the equation of motion of an
overdamped particle in a periodic potential, subject to a periodic force.
Numerical solutions of the Dirac equation confirm the existence of these
states, and demonstrate the chirality of states in their vicinity. Conductance
resonances are shown to accompany the emergence of these induced Dirac points.
