We present a joint analysis of the overlapping Balloon-borne Large Aperture Submillimetre Telescope (BLAST) 250, 350, 500 μm, and LABOCA 870 μm observations [from the LABOCA ECDFS Submm Survey (LESS) survey] of the Extended Chandra Deep Field-South. Out to z∼ 3, the BLAST filters sample near the peak wavelength of thermal far-infrared (FIR) emission from galaxies (rest-frame wavelengths ∼60–200 μm), primarily produced by dust heated through absorption in star-forming clouds. However, identifying counterparts to individual BLAST peaks is very challenging, given the large beams [full-width at half-maximum (FWHM) 36–60 arcsec]. In contrast, the ground-based 870 μm observations have a significantly smaller 19 arcsec FWHM beam, and are sensitive to higher redshifts (z∼ 1–5, and potentially beyond) due to the more favourable negative K-correction. We use the LESS data, as well as deep Spitzer and VLA imaging, to identify 118 individual sources that produce significant emission in the BLAST bands. We characterize the temperatures and FIR luminosities for a subset of 69 sources which have well-measured submillimetre (submm) spectral energy distributions (SEDs) and redshift measurements out to z∼ 3. For flux-limited sub-samples in each BLAST band, and a dust emissivity index β= 2.0, we find a median temperature T= 30 K (all bands) as well as median redshifts: z= 1.1 (interquartile range 0.2–1.9) for S250 > 40 mJy; z= 1.3 (interquartile range 0.6–2.1) for S350 > 30 mJy; and z= 1.6 (interquartile range 1.3–2.3) for S500 > 20 mJy. Taking into account the selection effects for our survey (a bias towards detecting lower-temperature galaxies), we find no evidence for evolution in the local FIR–temperature correlation out to z∼ 2.5. Comparing with star-forming galaxy SED templates, about 8 per cent of our sample appears to exhibit significant excesses in the radio and/or mid-IR, consistent with those sources harbouring active galactic nuclei (AGN). Since our statistical approach differs from most previous studies of submm galaxies, we describe the following techniques in two appendices: our ‘matched filter’ for identifying sources in the presence of point-source confusion; and our approach for identifying counterparts using likelihood ratios. This study is a direct precursor to future joint FIR/submm surveys, for which we outline a potential identification and SED measurement strategy.