Plasmonic metal nanostructures, in colloidal or surface-supported forms, have been extensively studied in the context of metamaterials design and applications, in particular as refractometric sensing platforms. Recently, high refractive index (high-n) dielectric subwavelength structures have been experimentally shown to support strong Mie scattering resonances, predicted to exhibit analogous refractive index sensing capabilities. Here we present the first experimental demonstration of the use of supported high-n dielectric nano/microparticle ensembles as refractive index sensing platforms, using cuprous oxide as a model high-n material. Single-crystalline CuO particles were deposited on transparent substrates using a chemical deposition scheme, showing well-defined electric and magnetic dipolar resonances (EDR and MDR, respectively) in the visible range, which change in intensity and wavelength upon changing the medium refractive index (n). The significant modulation of the MDR intensity when n is modified appears to be the most valuable empirical sensing parameter. The Mie scattering properties of CuO particles, particularly the spectral dependence of the MDR on n, are theoretically modeled to support the experimental observations. MDR extinction changes (i.e., refractive index sensitivity) per particle are >100 times higher compared to localized surface plasmon resonance (LSPR) changes in supported Au nanoislands, encouraging the evaluation of CuO and other high-n dielectric particles and sensing modes in order to improve the sensitivity in optical (bio)sensing applications.