jats:p We propose and analyze a quasirandom analogue of the classical push model for disseminating information in networks (“randomized rumor spreading”). In the classical model, in each round, each informed vertex chooses a neighbor at random and informs it, if it was not informed before. It is known that this simple protocol succeeds in spreading a rumor from one vertex to all others within jats:italicO</jats:italic> (log jats:italicn</jats:italic> ) rounds on complete graphs, hypercubes, random regular graphs, Erdős-Rényi random graphs, and Ramanujan graphs with probability 1 − jats:italico</jats:italic> (1). In the quasirandom model, we assume that each vertex has a (cyclic) list of its neighbors. Once informed, it starts at a random position on the list, but from then on informs its neighbors in the order of the list. Surprisingly, irrespective of the orders of the lists, the above-mentioned bounds still hold. In some cases, even better bounds than for the classical model can be shown. </jats:p>