We present a photosensitive three-hole microstructured optical fiber specifically designed to improve the refractive index sensitivity of a standard fiber Bragg grating (FBG) sensor photowritten in the suspended Ge-doped silica core. We describe the specific photowriting procedure used to realize gratings in such a fiber. We then determine their spectral sensitivity to the refractive index changes of material filling the holes surrounding the core. The sensitivity is compared with that of standard FBGs photowritten in a six-hole fiber with a larger core diameter. We demonstrate an improvement in the sensitivity by two orders of magnitude and reach a resolution of $3\times {10}^{-5}$ and $6\times {10}^{-6}$ around mean refractive index values of 1.33 and 1.40, respectively.