We introduce and demonstrate a simple spectrophotometer system insensitive to input polarization and with strong potential for compact and low-cost implementation. This technology has a wide variety of potential applications ranging from astronomy to medicine and even the cosmetics industry. To enable more powerful and portable microspectrometers we employ a novel design based on a tunable liquid crystal filter with polarization-independence, which is constructed of stacked liquid crystal polarization gratings (LCPGs). These switchable, anisotropic, thin diffraction gratings exhibit unique properties that include diffraction at visible and infrared wavelengths that can be coupled between only the zero- and first-orders (with nearly 100% and 0% experimentally verified efficiencies), depending on the applied voltage and wavelength of incident light. When combined with an elemental spatial filter, polarization-independent bandpass tuning can be achieved with minimum loss. Analogous to Lyot and Solc filters, several LCPGs are layered and introduced into a temporally resolved system using a single photodetector. The unique filter design enables improvement in terms of resolution and sensitivity by eliminating the polarization dependence present in all competing birefringence- based technologies. Also, the temporal detection system has a potential for improved miniaturization compared to any competing relevant approach and decreased cost by avoiding highly sensitive alignment, reflective diffraction components, Fabry-Perot cavities, and expensive detectors. In this work we describe the core principles of the tunable filter, present a representative spectrometer system design, report preliminary experimental data, and discuss the capabilities of the system in terms of spectral range, resolution, and sensitivity.