We propose microwave dielectric sensing by expoliting the highly sensitive frequency response of hyperbolically dispersive media. Through dispersion analysis, we show that the hyperbolic media are both spatially and spectrally sensitive to very slight changes in the constitutive dielectric material. We practically demonstrate the concept of microwave sensing in microstrip-based metallic grid structure cells fabricated on a rounded dielectric substrate . High dielectric sensitivities are shown when dielectric samples are placed in the path of the resonace cones, which are the characteristic high –field regions in hyperbolic medium under resonance. We demonstrate the resonant frequency shifts for the relative permittivities in the range 1-3.5