An original method using lock-in thermography with a laser excitation is proposed for the contactless estimation of open crack depths in metal with curvilinear shape. A continuous laser source regularly scans the structure under test leading to a periodical heating. The heat diffusion disturbances induced by a crack located in the thermal diffusion area are measured synchronously with the laser scans. The thermal signature of the crack is extracted from the amplitude of surface temperature images for various scanning speeds of the thermal source. The thermal signatures are analysed according to a length representative of the thermal diffusion length and to the radius curvature of the crack to give a local evaluation of the crack depth. The method, explained with 3D simulations, is experimentally implemented and tested with calibrated curvilinear cracks. The results demonstrate the potentiality of multi-speed laser lock-in thermography method as a contactless measurement tool for the evaluation of complex crack shapes up to 3 mm depth.