In the method, the wire (1) is driven in translation in at least one direction (F2) and, during its displacement in translation (F2), it penetrates into the piece to be cut (2) via a penetration face (20A) of said piece (2). According to the invention, a cushioning layer (9A) is positioned opposite the penetration face (20A) of the piece to be cut (2) such that, before penetrating into the piece (2), the wire (1) passes through the cushioning layer (9A). If the wire (1) is driven in translation in an alternating back-and-forth motion, and penetrates into the piece to be cut (2) in alternation via two opposite penetration faces (20A, 20B), two cushioning layers (9A, 9B) are respectively positioned opposite the two penetration faces.