Assumption-based argumentation (ABA) is one of the most-studied formalisms for structured argumentation. While ABA is a general formalism that can be instantiated with various different logics, most attention from the computational perspective has been focused on the logic programming (LP) instantiation of ABA. Going beyond the LP-instantiation, we develop an algorithmic approach to reasoning in the propositional default logic (DL) instantiation of ABA. Our approach is based on iterative applications of Boolean satisfiability (SAT) solvers as a natural choice for implementing derivations as entailment checks in DL. We instantiate the approach for deciding acceptance and for assumption-set enumeration in the DL-instantiation of ABA under several central argumentation semantics, and empirically evaluate an implementation of the approach.