We study the stellar populations of Type Ia supernova (SN Ia) host galaxies using Sloan Digital Sky Survey (SDSS)-II spectroscopy. The main focus is on the relationships of SN Ia properties with stellar velocity dispersion and the stellar population parameters age, metallicity and element abundance ratios. We concentrate on a sub-sample of 84 SNe Ia from the SDSS-II Supernova Survey and find that SALT2 stretch factor values show the strongest dependence on stellar population age. Hence, more luminous SNe Ia appear in younger stellar progenitor systems. No statistically significant trends in the Hubble residual with any of the stellar population parameters studied are found. Moreover, the method of photometric stellar mass derivation affects the Hubble residual-mass relationship. For an extended sample (247 objects), including SNe Ia with SDSS host galaxy photometry only, the Hubble residual-mass relationship behaves as a sloped step function. In the high-mass regime, probed by our host spectroscopy sample, this relationship is flat. Below a stellar mass of ~2x10^10^M_{sun}_, i.e. close to the evolutionary transition mass of low-redshift galaxies, the trend changes dramatically such that lower mass galaxies possess lower luminosity SNe Ia after light-curve corrections. The sloped step function of the Hubble residual-mass relationship should be accounted for when using stellar mass as a further parameter for minimizing the Hubble residuals.