In a search with the Parkes radio telescope of 56 unidentified Fermi-LAT gamma-ray sources, we have detected 11 millisecond pulsars (MSPs), 10 of them discoveries, of which five were reported in Kerr et al. (2012). We did not detect radio pulsations from another six pulsars now known in these sources. We describe the completed survey, which included multiple observations of many targets done to minimize the impact of interstellar scintillation, acceleration effects in binary systems, and eclipses. We consider that 23 of the 39 remaining sources may still be viable pulsar candidates. We present timing solutions and polarimetry for five of the MSPs, and gamma-ray pulsations for PSR J1903-7051 (pulsations for five others were reported in the second Fermi-LAT catalog of gamma-ray pulsars). Two of the new MSPs are isolated and five are in >1 d circular orbits with 0.2-0.3 Msun presumed white dwarf companions. PSR J0955-6150, in a 24 d orbit with a ~0.25 Msun companion but eccentricity of 0.11, belongs to a recently identified class of eccentric MSPs. PSR J1036-8317 is in an 8 hr binary with a >0.14 Msun companion that is probably a white dwarf. PSR J1946-5403 is in a 3 hr orbit with a >0.02 Msun companion with no evidence of radio eclipses.