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We develop a new inversion approach to construct a 3-D structural and shear-wave velocity model of the crust based on teleseismic P-to-S converted waves. The proposed approach does not require local earthquakes such as body wave tomography, nor a large aperture seismic network such as ambient noise tomography, but a three-component station network with spacing similar to the expected crustal thickness. The main features of the new method are: (1) a novel model parametrization with 3-D mesh nodes that are fixed in the horizontal directions but can flexibly vary vertically; (2) the implementation of both sharp velocity changes across discontinuities and smooth gradients; (3) an accurate ray propagator that respects Snell’s law in 3-D at any interface geometry. Model parameters are inverted using a stochastic method composed of simulated annealing followed by a pattern search algorithm. The first application is carried out over the Central Alps, where long-standing permanent and the temporary AlpArray Seismic Network stations provide an ideal coverage. For this study we invert 4 independent parameters, which are the Moho discontinuity depth, the Conrad discontinuity depth, the P-velocity change at the Conrad and the average Vp/Vs of the crust. The 3-D inversion results clearly image the roots of the Alpine orogen, including the Ivrea Geophysical Body. The lower crust's thickness appears fairly constant. Average crustal Vp/Vs ratios are relatively higher beneath the orogen, and a low-Vp/Vs area in the northern foreland seems to correlate with lower crustal earthquakes, which can be related to mechanical differences in rock properties, probably inherited. Our results are in agreement with those found by 3-D ambient noise tomography, though our method inherently performs better at localizing discontinuities. Future developments of this technique can incorporate joint inversions, as well as more efficient parameter space exploration.