We search for high-mass resonances decaying into Z boson pairs using data corresponding to 6  fb−1 collected by the CDF experiment in pp collisions at s√=1.96  TeV. The search is performed in three distinct final states: ZZ→ℓ+ℓ−ℓ+ℓ−, ZZ→ℓ+ℓ−νν, and ZZ→ℓ+ℓ−jj. For a Randall-Sundrum graviton G∗, the 95% CL upper limits on the production cross section times branching ratio to ZZ, σ(pp→G*→ZZ), vary between 0.26 pb and 0.045 pb in the mass range 300<1000  GeV/c2.