Several complex systems can be modeled as large networks in which the state of the nodes continuously evolves through interactions among neighboring nodes, forming a high-dimensional nonlinear dynamical system. One of the main challenges of Network science consists in predicting the impact of network topology and dynamics on the evolution of the states and, especially, on the emergence of collective phenomena, such as synchronization. We address this problem by proposing a Dynamics Approximate Reduction Technique (DART) that maps high-dimensional (complete) dynamics unto low-dimensional (reduced) dynamics while preserving the most salient features, both topological and dynamical, of the original system. DART generalizes recent approaches for dimension reduction by allowing the treatment of complex-valued dynamical variables, heterogeneities in the intrinsic properties of the nodes as well as modular networks with strongly interacting communities. Most importantly, we identify three major reduction procedures whose relative accuracy depends on whether the evolution of the states is mainly determined by the intrinsic dynamics, the degree sequence, or the adjacency matrix. We use phase synchronization of oscillator networks as a benchmark for our threefold method. We successfully predict the synchronization curves for three phase dynamics (Winfree, Kuramoto, theta) on the stochastic block model. Moreover, we obtain the bifurcations of the Kuramoto-Sakaguchi model on the mean stochastic block model with asymmetric blocks and we show numerically the existence of periphery chimera state on the two-star graph. This allows us to highlight the critical role played by the asymmetry of community sizes on the existence of chimera states. Finally, we systematically recover well-known analytical results on explosive synchronization by using DART for the Kuramoto-Sakaguchi model on the star graph. Our work provides a unifying framework for studying a vast class of dynamical systems on networks.