We investigate the effect of substrate thickness on the transmission bandwidth of on-chip terahertz-frequency-range planar Goubau lines both experimentally and theoretically. The bandwidth and frequency resolution are improved through substrate thinning and geometry modifications (reducing reflections arising from the THz photoconductive generators and detectors). We demonstrate that the enhanced bandwidth (2 THz) and resolution (3.75 GHz) allows this type of on-chip waveguide to be used for spectroscopic measurements of polycrystalline materials from cryogenic (4 K) to room temperature (292 K) by recording vibrational absorption spectra from overlaid samples of lactose monohydrate.