We present a measurement of the top-quark mass using a sample of $t\bar{t}$ events in 5.7 fb$^{-1}$ of integrated luminosity from $p\bar{p}$ collisions at the Fermilab Tevatron with $\sqrt{s} =$ 1.96 TeV and collected by the CDF II Detector. We select events having no identified charged leptons, large missing transverse energy, and four, five, or six jets with at least one jet tagged as coming from a $b$ quark. This analysis considers events from the semileptonic \ttbar decay channel, including events that contain tau leptons. The measurement is based on a multidimensional template method. We fit the data to signal templates of varying top-quark masses and background templates, and measure a top-quark mass of $\mtop = \gevcc{\measStatSyst{172.3}{2.4}{1.0}}$.