Dust grains play a key role in the physics of star-forming regions, even though they constitute only ∼1 per cent of the mass of the interstellar medium. The derivation of accurate dust parameters such as temperature (Td), emissivity spectral index (β) and column density requires broad-band continuum observations at far-infrared wavelengths. We present Herschel-Spectral and Photometric Imaging Receiver Array (SPIRE) Fourier Transform Spectrometer (FTS) measurements of three starless cores: L1521E, L1521F and L1689B, covering wavelengths between 194 and 671 μm. This paper is the first to use our recently updated SPIRE-FTS intensity calibration, yielding a direct match with SPIRE photometer measurements of extended sources. In addition, we carefully assess the validity of calibration schemes depending on-source extent and on the strength of background emission. The broad-band far-infrared spectra for all three sources peak near 250 μm. Our observations therefore provide much tighter constraints on the spectral energy distribution (SED) shape than measurements that do not probe the SED peak. The spectra are fitted using modified blackbody functions, allowing both Td and β to vary as free parameters. This yields Td of 9.8±0.2, 15.6±0.5 and 10.9±0.2 K and corresponding β of 2.6∓0.9, 0.8∓0.1 and 2.4∓0.8 for L1521E, L1521F and L1689B, respectively. The derived core masses are 1.0±0.1, 0.10±0.01 and 0.49±0.05 M⊙, respectively. The core mass/Jeans mass ratios for L1521E and L1689B exceed unity indicating that they are unstable to gravitational collapse, and thus pre-stellar cores. By comparison, the elevated temperature and gravitational stability of L1521F support previous arguments that this source is more evolved and likely a protostar.