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A realistic model of transport properties of zinc oxide varistors is constructed from two-dimensional Voronoi networks and studied via computer simulations. In agreement with experimental microcontact measurements made on individual junctions, the networks are assumed to contain randomly distributed microjunctions of two types: (1) electrically active with highly nonlinear current-voltage (I-V) characteristics and (2) ohmic, i.e., with linear I-V characteristics. Effects of the ohmic grain boundaries in the network are simulated for various concentrations and resistivities. Shapes of the simulated I-V characteristics and current dependence of the coefficient of nonlinearity of the network are in good agreement with those experimentally observed for thin varistor samples and in the measurements employing various surface electrode patterns. It is found that the breakdown voltage of the networks increases with the number of the ohmic grain boundaries, except when their resistivity is so low that it becomes comparable with that of the ZnO grains. The maximal value of the coefficient of nonlinearity of the network is shown to be insensitive to the presence of the ohmic grain boundaries, regardless of their resistivity and concentration.