The properties of a magnonic crystal are expected to be strongly influenced by the presence of a thermal gradient. We investigated the propagation of backward volume and surface magnetostatic spin-waves in a 1D magnonic crystal (MC) exposed to a continuous spatial temperature gradient. It is shown that the thermal gradient applied along the propagation direction leads to a frequency shift and a modification of the transmission characteristics of the spin-waves. The frequency shift is caused by a variation in saturation magnetization due to the change in absolute temperature. The altered transmission manifests itself in a broadening of MC band gaps and the corresponding narrowing of the MC passbands and is understood to be a result of a spatial transformation of the spin-waves wavelengths in a thermal gradient. Furthermore, the transmission characteristics of spin-waves in a thermal gradient have been verified by numerical calculations based on the approach of the transmission matrix. The results of the calculations demonstrate a good agreement with the experimentally measured data.