We derive photometric redshifts from 17-band optical to mid-infrared photometry of 78 robust radio, 24-μm and Spitzer IRAC counterparts to 72 of the 126 submillimetre galaxies (SMGs) selected at 870 μm by LABOCA observations in the Extended Chandra Deep Field-South (ECDF-S). We test the photometric redshifts of the SMGs against the extensive archival spectroscopy in the ECDF-S. The median photometric redshift of identified SMGs is z= 2.2 ± 0.1, the standard deviation is σz= 0.9 and we identify 11 (˜15 per cent) high-redshift (z≥ 3) SMGs. A statistical analysis of sources in the error circles of unidentified SMGs identifies a population of possible counterparts with a redshift distribution peaking at z= 2.5 ± 0.2, which likely comprises ˜60 per cent of the unidentified SMGs. This confirms that the bulk of the undetected SMGs are coeval with those detected in the radio/mid-infrared. We conclude that at most ˜15 per cent of all the SMGs are below the flux limits of our IRAC observations and thus may lie at z≳ 3 and hence at most ˜30 per cent of all SMGs have z≳ 3. We estimate that the full S870 μm > 4 mJy SMG population has a median redshift of 2.5 ± 0.5. In contrast to previous suggestions, we find no significant correlation between submillimetre flux and redshift. The median stellar mass of the SMGs derived from spectral energy distribution fitting is (9.1 ± 0.5) × 1010 M⊙ although we caution that the uncertainty in the star formation histories results in a factor of ˜5 uncertainty in these stellar masses. Using a single temperature modified blackbody fit with β= 1.5, the median characteristic dust temperature of SMGs is 37.4 ± 1.4 K. The infrared luminosity function shows that SMGs at z= 2-3 typically have higher far-infrared luminosities and luminosity density than those at z= 1-2. This is mirrored in the evolution of the star formation rate density (SFRD) for SMGs which peaks at z˜ 2. The maximum contribution of bright SMGs to the global SFRD (˜5 per cent for SMGs with S870 μm≳ 4 mJy or ˜50 per cent extrapolated to SMGs with S870 μm > 1 mJy) also occurs at z˜ 2.