Terahertz-range dielectric properties of the common polymers low-density polyethylene (LDPE), cyclic olefin/ethylene copolymer (TOPAS®), polyamide-6 (PA6), and polytetrafluoroethylene (PTFE or Teflon®) are characterized in the ultra-broadband frequency window 2-15 THz, using a THz time-domain spectrometer employing air-photonics for the generation and detection of single-cycle sub-50 fs THz transients. The time domain measurements provide direct access to both the absorption and refractive index spectra. The polymers LDPE and TOPAS® demonstrate negligible absorption and spectrally-flat refractive index across the entire spectroscopy window, revealing the high potential of these polymers for applications in THz photonics such as ultra-broadband polymer-based dielectric mirrors, waveguides, and fibers. Resonant high-frequency polar vibrational modes are observed and assigned in polymers PA6 and PTFE, and their dielectric functions in the complete frequency window 2-15 THz are theoretically reproduced. Our results demonstrate the potential of ultrabroadband air-photonics-based THz time domain spectroscopy as a valuable analytic tool for materials science.