This work addresses the problem of pattern analysis in networks consisting of delay-coupled identical Lur'e systems. We study a class of nonlinear systems, which, being isolated, are globally asymptotically stable. Assembling such systems into a network via time-delayed coupling may result in the change of network equilibrium stability under parameter variation in the coupling. In this work, we focus on cases where a Hopf bifurcation causes the change of stability of the network equilibrium and leads to the occurrence of oscillatory modes (patterns). Moreover, some of these patterns can co-exist for the same set of coupling parameters, which makes the analysis by means of common methods, such as the Lyapunov-Krasovskii method or the analysis of Poincaré maps, cumbersome. A numerically efficient algorithm, aiming at the computation of the oscillatory patterns occurring in such networks, is presented. Moreover, we show that our approach is able to deal with co-existing patterns, and both stable and unstable regimes can be simultaneously computed, which gives deep insight into the network dynamics. In order to illustrate the efficiency of the method, we present two examples in which the instability of the network equilibria is caused by a subcritical and a supercritical Hopf bifurcation. In addition, a bifurcation analysis of the subcritical case is performed in order to further explain the occurrence of the detected coexisting modes.