A new sensing approach utilizing Mie resonances in metamaterial arrays composed of dielectric resonators is proposed. These arrays were found to exhibit specific, extremely high-Q factor (up to 15,000) resonances at frequencies corresponding to the lower edge of the array second transmission band. The observed resonances possessed with features typical for Fano resonances (FRs), which were initially revealed in atomic processes and recently detected in macro-structures, where they resulted from interference between local resonances and a continuum of background waves. Our studies demonstrate that frequencies and strength of Fano-type resonances in all-dielectric arrays are defined by interaction between local Mie resonances and Fabry-Perot oscillations of Bloch eigenmodes that makes possible controlling the resonance responses by changing array arrangements. The opportunity for obtaining high-Q responses in compact arrays is investigated and promising designs for sensing the dielectric properties of analytes in the ambient are proposed.