As part of our programme to map the large-scale distribution of galaxies behind the southern Milky Way, we observed 314 opticallyselected, partially-obscured galaxies in the Zone of Avoidance (ZOA) in the Crux and Great Attractor (GA) regions. An additional 29 galaxies were observed in the Vela ZOA survey region (because of the small numbers they are not discussed any further). The observations were conducted with the Parkes 64 m (210 ft) radio telescope, in a single-pixel pointed mode, reaching an rms noise level of typically 2−6 mJy over the velocity search range of 400 < v < 10 500 km s −1 . A total of 162 galaxies were detected (plus 14 galaxies in the Vela region). The detection rate is slightly higher than for the Hydra/Antlia region (52% versus 45%) observed in the same way. This can be explained by the prominence of the GA overdensity in the survey regions, which leads to a relatively higher fraction of nearby galaxies. It is also evident from the quite narrow velocity distribution (largely confined to 3000−6000 km s −1 ) and deviates significantly from the expectation of a uniform galaxy distribution for the given sensitivity and velocity range. No systematic differences were found between detections and non-detections, in terms of latitude, foreground extinction, or environment, except for the very central part of the rich Norma cluster, where hardly any galaxies were detected. A detailed investigation of the H i content of the galaxies reveals strong H i deficiency at the core of the Norma cluster (within about a 0.4 Abell radius), similar to what has been found in the Coma cluster. The redshifts obtained by this observing technique result in a substantial reduction of the so-called redshift ZOA. This is obvious when analysing the large-scale structure of the new H i data in combination with data from other (optical) ZOA redshift surveys. The lower latitude detections provide further evidence of the extension of the Norma Wall, across the ZOA, in particular its bending towards the Cen-Crux clusters above the Galactic plane at slightly higher redshift, rather than a straight continuation towards the Centaurus clusters.