Temporal stream logic (TSL) extends LTL with updates and predicates over arbitrary function terms. This allows for specifying data-intensive systems for which LTL is not expressive enough. In the semantics of TSL, functions and predicates are left uninterpreted. In this paper, we extend TSL with first-order theories, enabling us to specify systems using interpreted functions and predicates such as incrementation or equality. We investigate the satisfiability problem of TSL modulo the standard underlying theory of uninterpreted functions as well as with respect to Presburger arithmetic and the theory of equality: For all three theories, TSL satisfiability is neither semi-decidable nor co-semi-decidable. Nevertheless, we identify three fragments of TSL for which the satisfiability problem is (semi-)decidable in the theory of uninterpreted functions. Despite the undecidability, we present an algorithm – which is not guaranteed to terminate – for checking the satisfiability of a TSL formula in the theory of uninterpreted functions and evaluate it: It scales well and is able to validate assumptions in a real-world system design.