In this study, the mobility of the threefold-symmetric Bricard linkage and its network are analyzed using screw theory. First, the screw motion equation of the linkage is derived. By applying the modified Grübler–Kutzbach criterion, we deduce that the degree of freedom (DOF) of the linkage is equal to 1. Then, we analyze the mechanical network constructed of threefold-symmetric Bricard linkages and provide its topological constraint graph. Using graph theory and screw theory, the constraint matrix of the mechanical network is obtained. Then, we solve the matrix rank via linear column transformation. Results show that the DOF of the mechanical network is equal to 1. The mobility analysis method of the mechanical network proposed in this study facilitates the solution of the constraint matrix rank and can be used as a reference for other mechanical networks constructed of single-loop linkages.