Many extensions of the Standard Model predict large numbers of additional unstable particles whose decays in the early universe are tightly constrained by observational data. For example, the decays of such particles can alter the ratios of light-element abundances, give rise to distortions in the cosmic microwave background, alter the ionization history of the universe, and contribute to the diffuse photon flux... Constraints on new physics from such considerations are typically derived for a single unstable particle species with a single well-defined mass and characteristic lifetime. In this paper, by contrast, we investigate the cosmological constraints on theories involving entire ensembles of decaying particles --- ensembles which span potentially broad ranges of masses and lifetimes. In addition to providing a detailed numerical analysis of these constraints, we also formulate a set of simple analytic approximations for these constraints which may be applied to generic ensembles of unstable particles which decay into electromagnetically-interacting final states. We then illustrate how these analytic approximations can be used to constrain a variety of toy scenarios for physics beyond the Standard Model. For ease of reference, we also compile our results in the form of a table which can be consulted independently of the rest of the paper. It is thus our hope that this work might serve as a useful reference for future model-builders concerned with cosmological constraints on decaying particles, regardless of the particular model under study. (read more)