We investigate the Hubbard model on the anisotropic triangular lattice with two hopping parameters
t and ${t}^{\ensuremath{'}}$ in different spatial directions, interpolating between decoupled chains (
t=0) and the isotropic triangular lattice ($t={t}^{\ensuremath{'}}$). Variational wave functions that include both Jastrow and backflow terms are used to compare spin-liquid and magnetic phases with different pitch vectors describing both collinear and coplanar (spiral) order. For relatively large values of the on-site interaction $U/{t}^{\ensuremath{'}}\ensuremath{\gtrsim}10$ and substantial frustration, i.e., $0.3\ensuremath{\lesssim}t/{t}^{\ensuremath{'}}\ensuremath{\lesssim}0.8$, the spin-liquid state is clearly favored over magnetic states. Spiral magnetic order is only stable in the vicinity of the isotropic point, while collinear order is obtained in a wide range of interchain hoppings from small to intermediate frustration.