We derive photometric redshifts from 17-band optical to mid-infrared photometry of 78 robust radio, 24-mu m and Spitzer IRAC counterparts to 72 of the 126 submillimetre galaxies (SMGs) selected at 870 mu m 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 sigma(z) = 0.9 and we identify 11 (similar to 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 similar to 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 similar to 15 per cent of all the SMGs are below the flux limits of our IRAC observations and thus may lie at z greater than or similar to 3 and hence at most similar to 30 per cent of all SMGs have z greater than or similar to 3. We estimate that the full S-870 mu m > 4mJy 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) x 10(10)M(circle dot) although we caution that the uncertainty in the star formation histories results in a factor of similar to 5 uncertainty in these stellarmasses. Using a single temperature modified blackbody fit with beta = 1.5, the median characteristic dust temperature of SMGs is 37.4 +/- 1.4K. 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 similar to 2. The maximum contribution of bright SMGs to the global SFRD (similar to 5 per cent for SMGs with S-870 mu m greater than or similar to 4mJy or similar to 50 per cent extrapolated to SMGs with S-870 mu m > 1mJy) also occurs at z similar to 2.