The invention relates to a method for engraving screened printing cylinders by means of an engraving member (4) in an electronic engraving machine. An engraving control signal (GS) controlling said engraving member (4) is generated by superimposing a visual signal (B), representing the tone values to be reproduced, and a periodic screen signal (R) in order to produce an engraving screen. The engraving member (4) engraves a series of cup-shapes in a rotating printing cylinder (1) that are arranged in the engraving screen. The engraving member (4) performs a forward advance movement in the direction of this axis of the printing cylinder (1) in order to engrave the surface of the cup-shapes. Each printing colour in each set of printing colours is assigned its own specific engraving screen which is chosen in such a way that the moiré effects that occur during printing with other colours are minimized. In order to minimize the disruptive moiré effects even further at least one engraving screen (24) is engraved on the printing cylinder (1) in an offset position in relation to the other engraving screens of a set of colours in the direction of the forward advance movement of the engraving member (4) and/or in the direction of the periphery of the printing cylinder.