We investigate the stability properties of strongly continuous semigroups generated by operators of the form A−BB*, where A is the generator of a contraction semigroup and B is a possibly unbounded operator. Such systems arise naturally in the study of hyperbolic partial differential equations with damping on the boundary or inside the spatial domain. As our main results we present general sufficient conditions for nonuniform stability of the semigroup generated by A − BB* in terms of selected observability-type conditions on the pair (B*, A). The core of our approach consists of deriving resolvent estimates for the generator expressed in terms of these observability properties. We apply the abstract results to obtain rates of energy decay in one-dimensional and two-dimensional wave equations, a damped fractional Klein–Gordon equation and a weakly damped beam equation.