We present an extension to the Adam Gibbs (AG) model to describe the segmental dynamics of miscible polymer blends with strong interactions. We studied the segmental dynamics of cross-linked styrene butadiene rubber(SBR)/butadiene rubber (BR) blends of different microstructure, chain-end functionalization, and composition both unfilled and filled with precipitated silica by means of broadband dielectric spectroscopy (BDS). Contrary to what is observed for athermal miscible blends, the dynamics shows only a single segmental relaxation process due to the strong intermolecular interaction given by the cross-links. The temperature dependence of the relaxation times has been described and analyzed within the framework of a modified AG approach that takes into account the strong interactions between blend components due to the presence of the cross-links. The accuracy of the proposed model facilitates a deepened understanding of the dynamics of polymer blend systems based on the dynamics of its neat components.