The desynchronization problems in oscillatory networks is considered. A new desynchronization notion is introduced and desynchronization conditions are provided. The desynchronization notion is formulated in terms of Yakubovich oscillatority of the auxiliary synchronization error system. As an example, the network of diffusively coupled FitzHugh-Nagumo systems with undirected graph is considered. The simple inequality guaranteeing network desynchronization is derived. The simulation results confirm the validity of the obtained analytical results.