We use a real-space renormalization group (RSRG) to study the low-temperature dynamics of kinetically constrained Ising chains (KCICs). We consider the cases of the Fredrickson-Andersen (FA) model, the East model, and the partially asymmetric KCIC. We show that the RSRG allows one to obtain in a unified manner the dynamical properties of these models near their zero-temperature critical points. These properties include the dynamic exponent, the growth of dynamical length scales, and the behavior of the excitation density near criticality. For the partially asymmetric chain, the RG predicts a crossover, on sufficiently large length and time scales, from East-like to FA-like behavior. Our results agree with the known results for KCICs obtained by other methods.