Summary: 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 \(c \geq c_F^+\) and no traveling waves for \(c < c_F^+\). We also show that the map \(F \mapsto c_F^+\) is not continuous for the \(L^\infty\) norm on \(F\). Assuming more regularity of \(F\) close to the unstable state 0, we show that \(c_F^+ \geq c_F^\ast\), where the velocity \(c_F^\ast\) can be computed from the linearization of the equation around the unstable state 0. In addition, 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 \(c_F^+ = c_F^\ast\). Finally, we provide an example in which \(c_F^+\) is negative.