Compliant linear-motion mechanisms are of great use in precision machines due to their excellent performances such as infinite resolution and low cost. The accuracy of the mechanisms is an important consideration for mechanical design in applications, especially in the case of large working load. Considering that COmpliant Rolling-contact Element (CORE) pivot is characterized with high bearing capacity, the paper adopts it as a building block to design a family of compliant linear-motion mechanisms for applications of heavy load. These mechanisms are achieved by replacing four rigid pivots in a parallel four-bar mechanism with CORE pivots, and the motion accuracy is improved by means of contacting surfaces design of four CORE pivots. First, structures of CORE pivot are introduced and five extended arrangements for bearing heavy load are presented. Meanwhile, motion for the CORE pivot is analyzed and preconditions for achieving a pure roll are discussed. Then, configuration of the compliant linear-motion mechanisms constructed by CORE pivots is obtained, and kinematics of the mechanisms is analyzed and parametric design condition for rectilinear motion is modeled. Based on the condition, detailed topological structures of the mechanisms are designed. Finally, motion simulations and experiment tests are implemented to verify accuracy of the proposed mechanisms. The results demonstrate that the mechanisms proposed in this paper are capable of offering a high-precision linear motion and providing a promising application prospect in precision machines.