We exploit ALMA 870um (345GHz) observations of submillimetre sources in the Extended Chandra Deep Field South to investigate the far-infrared properties of high-redshift submillimetre galaxies (SMGs). Using the precisely located 870um ALMA positions of 99 SMGs, together with 24um and radio imaging of this field, we deblend the Herschel/SPIRE imaging of this region to extract their far-infrared fluxes and colours. The median photometric redshifts for ALMA LESS (ALESS) SMGs which are detected in at least two SPIRE bands increases with wavelength of the peak in their SEDs, with z=2.3+/-0.2, 2.5+/-0.3 and 3.5+/-0.5 for the 250, 350 and 500-um peakers respectively. We find that 34 ALESS SMGs do not have a >3-sigma counterpart at 250, 350 or 500-um. These galaxies have a median photometric redshift of z=3.3+/-0.5, which is higher than the full ALESS SMG sample; z=2.5+/-0.2. Using the photometric redshifts together with the 250-870um photometry, we estimate the far-infrared luminosities and characteristic dust temperature of each SMG. The median infrared luminosity of the S_870um>2mJy SMGs is L_IR=(3.0+/-0.3)x10^{12}Lo(SFR=300+/-30Mo/yr). At a fixed luminosity, the characteristic dust temperature of these high-redshift SMGs is 2-3K lower than comparably luminous galaxies at z=0, reflecting the more extended star formation occurring in these systems. By extrapolating the 870um number counts to S_ 870um=1mJy, we show that the contribution of S_870um>1mJy SMGs to the cosmic star formation budget is 20% of the total over the redshift range z~1-4. We derive a median dust mass for these SMGs of M_d=(3.6+/-0.3)x10^8Mo and by adopting an appropriate gas-to-dust ratio, we estimate an average molecular mass of M_H2=(4.2+/-0.4)x10^{10}Mo. Finally, we use our estimates of the H2 masses to show that SMGs with S_870um>1mJy contain ~10% of the z~2 volume-averaged H2 mass density at this epoch. ; Comment: Accepted for publication in MNRAS. 24 pages, 12 figures