A new class of frequency-selective surface filters (FSS) for terahertz (THz) applications is proposed and investigated both numerically and experimentally. A periodic FSS array of cross-shaped apertures is patterned on aluminum, deposited on thin foils of the low-loss cyclo-olefin polymer Zeonor. Apart from the fundamental filtering response of the FSS elements, we also observe very narrow-linewidth peaks with high transmittance, associated with guided-mode resonances in the dielectric substrate. The effect of the filter's geometrical parameters on its performance is systematically studied via finite-element method simulation and confirmed by time-domain spectroscopy characterization of the fabricated samples. Finally, thanks to the flexibility of the employed substrates, THz-FSS filters are also characterized in bent configuration, revealing a robust response in terms of the fundamental FSS passband filter and a high sensitivity of the GMR peaks. These features can be exploited in the design of novel THz filters or sensors.