  We explore the effect of the third nearest-neighbors on the magnetic
properties of the Heisenberg model on an anisotropic triangular lattice. We
obtain the phase diagram of the model using Schwinger-boson mean-field theory.
Competition between N\'eel, spiral and collinear magnetically ordered phases is
found as we vary the on the ratios of the nearest, J1, next-nearest, J2, and
third-nearest, J_3, neighbor exchange couplings. A spin liquid phase is
stabilized between the spiral and collinear ordered states when J2/J1 < 1.8 for
rather small J3/J1 < 0.1. The lowest energy two-spinon dispersions relevant to
neutron scattering experiments are analyzed and compared to semiclassical
magnon dispersions finding significant differences in the spiral and collinear
phases between the two approaches. The results are discussed in the context of
the anisotropic triangular materials: Cs2CuCl4 and Cs2CuBr4 and layered organic
materials, kappa-(BEDT-TTF)2X and Y[Pd(dmit)2]2.
