This paper advocates Riemannian multi-manifold modeling for network-wide time-series analysis: Dynamic brain-network data yield features which are viewed as points in or close to a union of a finite number of submanifolds of a Riemannian manifold. Distinguishing disparate time series amounts then to clustering multiple Riemannian submanifolds. To this end, two feature-generation schemes for network-wide dynamic time series are put forth. The first one is motivated by Granger-causality arguments and uses an auto-regressive moving average model to map low-rank linear vector subspaces, spanned by column vectors of observability matrices, to points into the Grassmann manifold. The second one utilizes (non-linear) dependencies among network nodes by introducing kernel-based partial correlations to generate points in the manifold of positive-definite matrices. Capitalizing on recently developed research on Riemannian-submanifold clustering, an algorithm is provided to differentiate time series based on their Riemannian-geometry properties. Extensive numerical tests on synthetic and real fMRI data demonstrate that the proposed framework outperforms classical and state-of-the-art techniques in clustering brain-network states/structures.