This paper studies fictitious play in networks of noncooperative two-player games. We show that continuous-time fictitious play converges to Nash equilibrium provided that the overall game is zero-sum. Moreover, the rate of convergence is 1/T , regardless of the size of the network. In contrast, arbitrary n-player zero-sum games do not possess the fictitious-play property. As an extension, we consider networks in which each bilateral game is strategically zero-sum, a weighted potential game, or a two-by-two game. In those cases, convergence requires either a condition on bilateral payoffs or that the underlying network structure is acyclic. The results are shown to hold also for the discrete-time variant of fictitious play, which entails a generalization of Robinson's theorem to arbitrary zero-sum networks. Applications include security games, conflict networks, and decentralized wireless channel selection.