Because of remarkable characteristics such as superior speeds and accelerations, high stiffness and good dynamic performance, parallel robots are being increasingly adjusted to different task requirements in the manufacturing field. Their parallel structures made by closed-loop kinematic chains are better suited to develop new curved and multidirectional fabrication strategies in Additive Manufacturing. Based on this application, the conceptual design and dimensional optimization of a new structure of the linear delta parallel robot for Additive Manufacturing (three-dimensional printing) is presented. The new structure uses an innovative concept of delta mechanism with single legs and rotational joints, which consists of 12 links (three single parallel legs), three prismatic joints, and 11 revolute joints. A particular feature of the proposed mechanism is that it contains a joint common to all the kinematic chains instead of a mobile platform. Quality function deployment is used as a methodology for conceptual design. Then the kinematics of the mechanism is described in detail, including mobility analysis, inverse and direct kinematics, and a study of dimensional optimization. A method of efficient optimization based on genetic algorithms is used to find the minimum dimensional parameters of the robot, considering the maximization of the useful workspace as main performance index. Finally, a prototype of the robot is developed to validate the design concepts and functionality of the machine.