We consider general fully nonlinear discrete reaction-diffusion equations u_t = F [u], described by some function F. In the positively monostable case, we study monotone traveling waves of velocity c, connecting the unstable state 0 to a stable state 1. Under Lipschitz regularity of F , we show that there is a minimal velocity c ^+_F such that there is a branch of traveling waves with velocities c ≥ c^+_F , and no traveling waves for c < c^+_F. We also show that the map F → c^+_F is not continuous for the L ^∞ norm on F. Assuming more regularity of F close to the unstable state 0, we show that c^+_F ≥ c^*_F where the velocity c^*_F can be computed from the linearization of the equation around the unstable state 0. We show that the inequality can be strict for certain nonlinearities F. On the contrary, under a KPP condition on F , we show the equality c^+_F = c^*_F. Finally, we also give an example where c^+_F is negative.