A method for printing continuous flax fiber-reinforced plastic (CFFRP) composite parts by five-axis three-dimensional (3D) printer, based on fused filament fabrication (FFF) technology, has been developed. FFF printed parts usually need supporting structures, have a stair step effect, and unfavorable mechanical properties. In order to address these deficiencies, continuous natural fiber prepreg filaments were first manufactured, followed by curved path planning for the model for generation of the G-code, and finally printed by a five-axis 3D printer. The surface quality of printed parts was greatly improved. The tensile strength and modulus of CFFRP increased by 89% and 73%, respectively, compared with polylactic acid (PLA) filaments. The flexural strength and modulus of the 3D-printed CFFRP specimens increased by 211% and 224%, respectively, compared with PLA specimens. The maximal curved bending force load and stiffness of the 3D-printed CFFRP specimens increased by 39% and 115%, respectively, compared with the flat slicing method. Advanced light structures, such as leaf springs, can be designed and manufactured by taking advantage of the favorable properties of these composites, which endow them with significant potential for application in the field of automobiles.