We present here a SAT-based framework for LTLf (Linear Temporal Logic on Finite Traces) satisfiability checking. We use propositional SAT-solving techniques to construct a transition system for the input LTLf formula; satisfiability checking is then reduced to a path-search problem over this transition system. Furthermore, we introduce CDLSC, an improved algorithm that leverages information produced by propositional SAT solvers from both satisfiability and unsatisfiability results. Experimental evaluation show that our specialized CDLSC (Conflict-Driven LTLf Satisfiability Checking) algorithm for LTLf satisfiability checking outperforms approaches based on the state-of-the-art SAT-based LTL satisfiability-checking techniques, demonstrating approximately four-fold speed-up in total running time.