This article addresses the path synthesis of RCCC (revolute-cylindrical-cylindrical-cylindrical) linkages, a problem that has not received due attention in the literature. Compared with planar and spherical four-bar linkages, a RCCC linkage has many more design parameters, which lead to a complex formulation of the path synthesis problem and, consequently, to a quite challenging system of algebraic equations. In this article, the problem is solved with a novel formulation of path synthesis for visiting a number of prescribed positions. This is achieved by means of an alternative coordinate system, which allows point coordinates to be expressed with the aid of two vectors fixed to the same body. By this means, the rotation matrix used to represent the coupler link attitude is obviated. The synthesis equations are then formulated in a simple form. Our formulation confirms that path synthesis admits exact solutions for up to nine prescribed positions, which proves a landmark claim submitted by Burmester. Examples are included to demonstrate the path synthesis procedure with the method thus developed.