Based on this paper, ZnO wires give Ohmic (linear current-voltage relationship) behaviour when the electrical contacts between the wire and supply are properly made:
This is an interesting question. The answer depends on the presence of defects in the ZnO crystal. Most of the surface defects acts as leaky paths for the currents in a Schottky junction. Therefore, one may not get good rectifying current nature. These defects are also responsible for barrier inhomogeneities at the interfaces which affects the ideality factor, barrier height and richardson constant of ZnO.