Vibrations of small (microliter) sessile liquid droplets were studied using a simple optical deflection technique. The droplets were made to elongate in one direction by taking advantage of the anisotropic wetting of the liquids on structured diffraction grating surfaces. They were vibrated by applying a puff of nitrogen gas. Motion of the droplets was monitored by scattering laser light from their surfaces. The scattered light was collected using a photodiode, and the resulting time-dependent intensity signals were Fourier-transformed to obtain the vibrational response of the drops. The vibrational spectra of elongated sessile drops were observed to contain two closely spaced peaks. A simple model that considers the frequency of capillary wave fluctuations on the surfaces of the drops was used to show that the vibrational frequencies of these peaks correspond to standing wave states that exist along the major and minor profile lengths of the droplets.