Despite a rich and interdisciplinary literature on time-series analysis, the brain's complex distributed dynamics are typically quantified using only a limited set of manually selected representations and statistics. This leaves open the possibility that alternative dynamical properties may outperform those reported for a given application. Here we address this limitation by introducing a systematic procedure to compare diverse, interpretable features of functional magnetic resonance imaging (fMRI) data, drawn from comprehensive algorithmic libraries to capture both dynamical properties of individual brain regions (intra-regional activity) and patterns of statistical dependence between regions (inter-regional functional coupling). We apply our data-driven approach to investigate alterations to dynamical structures in resting-state fMRI in participants with schizophrenia (SCZ), bipolar 1 disorder (BP), attention-deficit hyperactivity disorder (ADHD), and autism spectrum disorder (ASD). Our findings broadly support the use of linear time-series analysis techniques for fMRI-based case--control analysis, while also identifying novel informative dynamical structures that have previously received less attention. While some simple statistical representations of fMRI dynamics perform surprisingly well (e.g., properties within a single brain region), we found performance improvements through combining intra-regional properties with inter-regional coupling---consistent with distributed, multifaceted changes to fMRI dynamics in neuropsychiatric disorders. The data-driven approach here enables the systematic identification and interpretation of disorder-specific dynamical signatures, with applicability beyond neuroimaging to diverse scientific problems involving complex time-varying systems.