We demonstrate a narrow-bandwidth acoustooptic tunable filter with very low polarization-sensitivity using a dispersion-compensating fiber that has a large group index difference between the core and the cladding modes with a specific stress distribution. The 3-dB bandwidth of the filter is 0.8 nm with the interaction length of 10 cm, and the polarization-dependent center-wavelength splitting is 0.04 nm at the wavelength around 1550 nm. The origin of the low polarization-dependence is successfully explained through the analysis on the stress field in the fiber.