We propose a scheme to study exactly the kinetic Ising model on fractals under the linear approximation. This enables us to obtain the critical slowing-down behavior of the longest relaxation time of the kinetic Ising model on a fractal exactly. As a result, we show that there are only two time scales in the relaxation process of the kinetic Ising model on a fractal cluster under the linear approximation; the critical dynamic exponent is always a constant that is unrelated to the temperature of the system and is equal to 2. The implication of our results to the linear approximation of the critical dynamics of the kinetic Ising model on fractals as well as to the time-dependent renormalization-group method is discussed.