We have developed long-period fiber gratings (LPFGs) utilizing the photoelastic effect and have demonstrated polarization-independent operation. The LPFG is made by pressing a standard, jacketed single-mode fiber between a flat plate and a plate with grooves mechanically machined with a suitable period. The grating's transmission spectrum is easily tuned by adjusting pressure, grating tilt, and length. Furthermore, the grating can be completely erased by removing the pressure from the fiber. Grating attenuation greater than 25 dB has been demonstrated with a notch-location polarization dependence of +-4 nm. In this paper we report drastic reduction in this polarization dependence by two different approaches. Passing through the grating a second time after reflecting off a Faraday rotator mirror was successful; this method may be used with other types of LPFGs. The second approach utilizes our mechanical grating's ability to be double-passed with two fibers side-by-side. Between passes, a fiber-loop half-wave plate aligned at 45 degrees to the plane of the grooved plate swaps power between x- and y-polarization states. The resulting output's measured polarization dependence was smaller than +/- 0.2 nm. Further improvement is expected through careful tuning of the wave plate. We also report a computer model of the filter spectrum and its polarization dependence, which takes into account non-uniform index perturbation, lossy cladding modes, cladding index perturbation, as well as the polarization dependence of the photoelastic effect, characteristics not usually present in UV-induced LPFGs. The model generates transmission spectra that agree quite well with experimental results.