Numerically-exact values of the mean walk length before first encounter for reactants diffusing on a lattice are calculated by generalizing a Markovian method introduced previously. The approach developed is illustrated by presenting results for two random walkers diffusing simultaneously on a square-planar lattice; the results obtained are compared with those calculated assuming one of the reactant partners is stationary. In almost all cases, it is found that the former process is the more efficient. Analytic arguments are presented and Monte Carlo studies are carried out to support the results obtained.