We exploit Atacama Large Millimeter Array (ALMA) 870 μm observations of sub-millimetre sources in the Extended Chandra Deep Field South to investigate the far-infrared properties of high-redshift sub-millimetre galaxies (SMGs). Using the precisely located 870 μm ALMA positions of 99 SMGs, together with 24μm and radio imaging, we deblend the Herschel/SPIRE imaging to extract their far-infrared fluxes and colours. The median redshifts for ALMA LESS (ALESS) SMGs which are detected in at least two SPIRE bands increases with wavelength of the peak in their spectral energy distributions (SEDs), with z = 2.3 ± 0.2, 2.5 ± 0.3 and 3.5 ± 0.5 for the 250, 350 and 500 μm peakers, respectively. 34 ALESS SMGs do not have a >3σ counterpart at 250, 350 or 500 μm. These galaxies have a median photometric redshift derived from the rest-frame UV-mid-infrared SEDs of z = 3.3 ± 0.5, which is higher than the full ALESS SMG sample; z = 2.5 ± 0.2. We estimate the far-infrared luminosities and characteristic dust temperature of each SMG, deriving LIR = (3.0 ± 0.3) × 1012 L⊙ (SFR = 300 ± 30 M⊙ yr-1) and Td = 32 ± 1 K. The characteristic dust temperature of these high-redshift SMGs is ΔTd = 3-5 K lower than comparably luminous galaxies at z = 0, reflecting the more extended star formation in these systems. We show that the contribution of S870 μm ≥ 1 mJy SMGs to the cosmic star formation budget is 20 per cent of the total over the redshift range z ˜ 1-4. Adopting an appropriate gas-to-dust ratio, we estimate a typical molecular mass of the ALESS SMGs of M_H_2 = (4.2 ± 0.4) × 1010 M⊙. Finally, we show that SMGs with S870 μm > 1 mJy (LIR ≳ 1012 L⊙) contain ˜ 10 per cent of the z ˜ 2 volume-averaged H2 mass density.