The parasitic motion of a parallel four-bar mechanism (PFBM) is undesirable for designers. In this paper, the rigid joints in PFBM are replaced with their flexural counterparts, and the center shift of rotational flexural pivots can be made full use of in order to compensate for this parasitic motion. First, three schemes are proposed to design a family of ultraprecision linear-motion mechanisms. Therefore, the generalized cross-spring pivots are utilized as joints, and six configurations are obtained. Then, for parasitic motion of these configurations, the compensation condition is presented, and the design space of geometric parameters is given. Moreover, the characteristic evaluation of these configurations is implemented, and an approach to improve their performances is further proposed. In addition, a model is developed to parametrically predict the parasitic motion and primary motion. Finally, the analytic model is verified by finite element analysis (FEA), so these linear-motion mechanisms can be employed in precision engineering.