The critical dynamics of the kinetic Glauber-Ising model is studied on a family of the diamond-type hierarchical lattices with various branches. By carrying out the time-dependent real-space renormalizationgroup transformation to the master equation of the systems considered, the dynamic exponent is calculated. We find that the dynamic exponent depends on fractal dimension df or the branch number m in a generator, and that it increases with the increase of df or m. We notice that for the case of m51 one-dimensional spin chain, df51) our result z52 is the same as the exact result obtained by Glauber, and for the case of m52 the simplest one in the diamond-type hierarchical lattices, df52) the exponent z52.626 is higher than those of the two-dimensional regular lattice and the triangular lattice.