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We study the quench dynamics of the three-dimensional Kitaev (spin) model under a linear drive using both exact numerical calculations and analytical "independent crossing approximation". Unlike the two-dimensional Kitaev model, the three-dimensional Kitaev model reduces to a multilevel Landau-Zener problem for each momentum. We show that for the slow quench, the defect density is proportional to the quench rate $1/\tau$. We find that the zeros of the relevant coupling between the levels determine the non-adiabatic condition for the production of defects. The contour on which the energy spectrum becomes gapless does not play an active role. The asymptotic behavior of the defect density crucially depends on the way the system reaches the non-adiabatic regime during the quenching process. We analytically show that defect correlation varies as $\tau^{-1} e^{-A/\tau}$, where $A$ is a constant independent of $\tau$. For the slow quench, the qualitative dependence of the entropy (produced during the quenching process) on the quench time is the same as that of the defect correlation, indicating a close connection between the defect correlation and the entropy content of the final state. Possible experimental realization of such quench dynamics is also described briefly., Comment: Author sequence changed, version accepted in Phys. Rev. B