The density of the complex temperature zeros of the partition function of the two-dimensional Ising model on completely anisotropic triangular lattices is investigated near real and complex critical points. The non-uniform behaviour of the density of zeros at interior and boundary critical points is studied analytically, and numerically for a lattice with interactions in the ratios 3:2:1. The limiting behaviour of the density at complex critical points depends on the direction of approach in the complex plane. In the neighbourhood of interior critical points one finds only a single layer of zeros. But generally there are two layers or superposed sets of zeros with different distributions, and different limiting densities at boundary critical points. On anisotropic quadratic and on partially anisotropic triangular lattices the two layers become identical, by symmetry of the partition function. The divergence of the (two- dimensional) density of zeros at real critical points is discussed briefly in relation to scaling theory.