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We prove that any Markov chain that performs local, reversible updates on randomly chosen vertices of a bounded-degree graph necessarily has mixing time at least Ω(n log n), where it is the number of vertices. Our bound applies to the so-called "Glauber dynamics" that has been used extensively in algorithms for the Ising model, independent sets, graph colorings and other structures in computer science and statistical physics, and demonstrates that many of these algorithms are optimal up to constant factors within their class. Previously no super-linear lower bound for this class of algorithms was known. Though widely conjectured, such a bound had been proved previously only in very restricted circumstances, such as for the empty graph and the path. We also show that the assumption of bounded degree is necessary by giving a family of dynamics on graphs of unbounded degree with mixing time O(n).