Metamaterials with hyperbolic dispersion based on metallic nanorod arrays provide a flexible platform for the design of bio- and chemical sensors and nonlinear devices, allowing the incorporation of functional materials into and onto the plasmonic metamaterial. Here, we have investigated, both analytically and numerically, the dependence of the optical response of these metamaterials on refractive index variations in commonly used experimental sensing configurations, including transmission, reflection, and total internal reflection. The strategy for maximising refractive index sensitivity for different configurations has been considered, taking into account contributions from the superstrate, embedding matrix, and the metal itself. It is shown that the sensitivity to the refractive index variations of the host medium is at least 2 orders of magnitude higher than to the ones originating from the superstrate. It is also shown that the refractive index sensitivity increases for higher-order unbound and leaky modes of the metamaterial sensor. The impact of the transducer's thickness was also analysed showing significant increase of the sensitivity for the thinner metamaterial layers (down to few 0.01 fraction of wavelength and, thus, requiring less analyte) as long as modes are supported by the structure. In certain configurations, both TE and TM-modes of the metamaterial transducer have comparable sensitivities. The results provide the basis for the design of new ultrasensitive chemical and biosensors outperforming both surface plasmon polaritons and localised surface plasmons based transducers.