We report on a search for anomalous production of Z boson pairs through a massive resonance decay in data corresponding to 2.5–2.9  fb−1 of integrated luminosity in pp collisions at s√=1.96  TeV using the CDF II detector at the Fermilab Tevatron. This analysis, with more data and channels where the Z bosons decay to muons or jets, supersedes the 1.1  fb−1 four-electron channel result previously published by CDF. In order to maintain high efficiency for muons, we use a new forward tracking algorithm and muon identification requirements optimized for these high signal-to-background channels. Predicting the dominant backgrounds in each channel entirely from sideband data samples, we observe four-body invariant mass spectra above 300  GeV/c2 that are consistent with background. We set limits using the acceptance for a massive graviton resonance that are 7–20 times stronger than the previously published direct limits on resonant ZZ diboson production.