Artificial spin ices are ensembles of geometrically-arranged, interacting nanomagnets which have shown promising potential for the realization of reconfigurable magnonic crystals. Such systems allow for manipulating spin waves on the nanoscale and potentially using them as information carriers. However, there are two general obstacles to the realization of spin ice-based magnonic crystals: the magnetic state of spin ices is difficult to reconfigure and the magnetostatic interactions between the islands are often weak, preventing mode coupling. We demonstrate, using micromagnetic modeling, that coupling a reconfigurable spin ice geometry made of weakly interacting nanomagnets to a soft magnetic underlayer creates a complex system exhibiting strongly coupled modes. These give rise to spin wave channels, which can be reconfigured by the magnetic state of the spin ice. These findings open the door to the realization of reconfigurable magnonic crystals with potential applications for data transport and processing in magnonic-based logic architectures.