Recently, the first experimental observation of a new class of non-diffracting optical beams that freely accelerate in space was reported. These so-called Airy beams were shown to be useful for optical micro-manipulation of small particles and for the generation of curved plasma channels in air. To date, these beams have been generated only by using linear diffractive elements. Here, we show a new way of generating Airy beams by using three-wave mixing processes, which occur in asymmetric nonlinear photonic crystals. We experimentally generated a second-harmonic Airy beam and examined the tuning properties of the nonlinear interaction and propagation dynamics of the pump and second-harmonic output beams. This nonlinear generation process enables Airy beams to be obtained at new wavelengths, and opens up new possibilities for all-optical switching and manipulation of Airy beams.