[abridged] We derive photometric redshifts from 17-band optical to mid-IR photometry of 74 robust counterparts to 68 of the 126 submillimetre galaxies (SMGs) selected at 870um by LABOCA observations in the ECDFS. The median photometric redshift of identified SMGs is z=2.2pm0.1, the interquartile range is z=1.8-2.7 and we identify 10 (~15%) high-redshift (z>3) SMGs. We derive a simple redshift estimator for SMGs based on the 3.6 and 8um fluxes, which is accurate to Delta_z~0.4 for SMGs at z<4. A statistical analysis of sources around unidentified SMGs identifies a population of likely counterparts with a redshift distribution peaking at z=2.5pm0.3, which likely comprises ~60% of the unidentified SMGs. This confirms that the bulk of the undetected SMGs are coeval with those detected in the radio/mid-IR. We conclude that ~15% of all the SMGs are below the flux limits of our survey and lie at z>3 and hence ~30% of all SMGs have z>3. We estimate that the full S_870um>4mJy SMG population has a median redshift of 2.5pm0.6. In contrast to previous suggestions we find no significant correlation between S_870um and redshift. The median stellar mass of the SMGs derived from SED fitting is (9.2pm0.9)x10^10Msun and the interquartile range is (4.7-14)x10^10Msun, although we caution that uncertainty in the star-formation histories results in a factor of ~5 uncertainty in these stellar masses. The median characteristic dust temperature of SMGs is 35.9pm1.4K and the interquartile range is 28.5-43.3K. The infrared luminosity function shows that SMGs at z=2-3 typically have higher far-IR 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% for SMGs with S_870um>4mJy; ~50% for SMGs with S_870um>1mJy) also occurs at z~2.
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