Sheets are conveyed through a rotary press with linear travel grippers. The novel conveying apparatus prevents collisions between the grippers and printing unit cylinders. The grippers are fixed to circulating links to convey a sheet through a nip between two cylinders that interact during printing. The links and the cylinders are coupled to drives that can be activated separately. In each case a collision disk is mounted on a shaft journal of at least one cylinder. A slip clutch is provided between the collision disk and the cylinder. A braking element is coupled to the grippers which, in the event of non-synchronous running with the cylinder, runs onto the collision disk and prevents the collision disk from moving.