This paper considers a generic approach to runtime verification of temporal properties over firstorder theories. This allows especially for the verification of multi-threaded, object-oriented systems. It presents a general framework lifting monitor synthesis procedures for propositional temporal logics to a temporal logic over structures within some first-order theory. To evaluate such specifications SMT solving and classical monitoring of propositional temporal properties are combined. The monitoring procedure was implemented for linear-time temporal logic (LTL) based on the Z3 SMT solver and evaluated regarding runtime performance.