The contribution of alloy scattering to the electronic scattering rate at the Brillouin-zone center in 18 III-V ternary semiconducting alloys is calculated in the Born approximation using the semi-empirical tight-binding parameters of Vogl, Hjalmarson, and Dow. Three types of disorder for a ternary alloy of the type AxB1-xC are included in these calculations. These are (1) ordinary diagonal disorder or disorder in the diagonal tight-binding energies associated with the constituents A and B, (2) off-diagonal disorder or disorder in the tight-binding transfer energies between the constituents A and C and the constituents B and C, and (3) redistribution disorder which takes into account the fact that the diagonal tight-binding energy of constituent C is different in substance AC than in substance BC. The latter two types of disorder are taken into account for the first time by our calculations, and we show that all three types of disorder can make roughly equal and sometimes competing contributions to the alloy scattering rate. Scattering rates appropriate for both conduction-band and valence-band scattering are presented and chemical trends in these rates for the alloys considered are predicted.