A search for flavor-changing neutral currents in top-quark decays $t\ensuremath{\rightarrow}Zq$ is performed in events produced from the decay chain $t\overline{t}\ensuremath{\rightarrow}Zq+Wb$, where both vector bosons decay leptonically, producing a final state with three leptons (electrons or muons). A data set collected with the CMS detector at the LHC is used, corresponding to an integrated luminosity of $19.7\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of proton-proton collisions at a center-of-mass energy of 8 TeV. No excess is seen in the observed number of events relative to the standard model prediction; thus, no evidence for flavor-changing neutral currents in top-quark decays is found. A combination with a previous search at 7 TeV excludes a $t\ensuremath{\rightarrow}Zq$ branching fraction greater than 0.05% at the 95% confidence level.