A simple kinetic model of the weakly coupled linear Ising chains is treated. The $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$-dependent dynamical susceptibility is obtained, introducing the interchain molecular field into exact one-dimensional solutions. The usual molecular-field-type critical slowing down as $T\ensuremath{\rightarrow}{T}_{c}$ is accompanied by an additional noncritical, highly anisotropic slowing down due to high-intrachain short-range order. For the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}=0$ mode the factor of the one-dimensional slowing down goes with decreasing temperature as the square of the one-dimensional correlation length. In the case of a strong anisotropy (${J}_{\ensuremath{\parallel}}\ensuremath{\gg}{J}_{\ensuremath{\perp}}$), this factor surpasses the value of the ratio $\frac{{J}_{\ensuremath{\parallel}}}{{J}_{\ensuremath{\perp}}}$ already well above Tc.