The automata-theoretic approach to verification of LTL relies on an algorithm for finding accepting cycles in the product of the system and a Büchi automaton for the negation of the formula. Explicit-state model checkers typically construct the product space "on the fly" and explore the states using depth-first search. We survey algorithms proposed for this purpose and propose two improved algorithms, one based on nested DFS, the other on strongly connected components. We compare these algorithms both theoretically and experimentally and determine cases where both algorithms can be useful.