We present a search for new phenomena in events with two reconstructed Z bosons and large missing transverse momentum, sensitive to processes pp→X2X2→ZZX1X1, where X2 is an unstable particle decaying as X2→ZX1 and X1 is undetected. The particles X1 and X2 may be, among other possibilities, fourth-generation neutrinos or supersymmetric particles. We study the final state in which one Z boson decays to two charged leptons and the second decays hadronically. In data corresponding to an integrated luminosity of 4.2  fb−1 from proton-antiproton collisions recorded by the CDF II detector at the Tevatron, with center-of-mass energy of 1.96 TeV, we find agreement between data and standard-model backgrounds. We calculate 95% confidence level upper limits on the cross section of the process pp→X2X2→ZZX1X1 ranging from 50 fb to 1 pb, depending on the masses of X1 and X2.