An orientation-sensitive fiber-optic accelerometer based on grating inscription over fiber cladding has been demonstrated. The sensor probe comprises a compact structure in which a short section of thin-core fiber (TCF) stub containing a "cladding" fiber Bragg grating (FBG) is spliced to another single-mode fiber (SMF) without any lateral offset. A femtosecond laser side-illumination technique was utilized to ensure that the grating inscription remains close to the core-cladding interface of the TCF. The core mode and the cladding mode of the TCF are coupled at the core-mismatch junction, and two well-defined resonances in reflection appear from the downstream FBG, in which the cladding resonance exhibits a strong polarization and bending dependence due to the asymmetrical distribution of the cladding FBG along the fiber cross section. Strong orientation dependence of the vibration (acceleration) measurement has been achieved by power detection of the cladding resonance. Meanwhile, the unwanted power fluctuations and temperature perturbations can be referenced out by monitoring the fundamental core resonance.