Snake-inspired locomotion is much more maneuverable compared to conventional locomotion concepts and it enables a robot to navigate through rough terrain. A rectilinear gait is quite flexible and has the following benefits: functionality on a wide variety of terrains, enables a highly stable robot platform, and provides pure undulatory motion without passive wheels. These benefits make rectilinear gaits especially suitable for search and rescue applications. However, previous robot designs utilizing rectilinear gaits were slow in speed. This paper introduces a new class of rectilinear gaits to be utilized by a snake-inspired robot design which is capable of pure linear motion and variable traction. The general model for the gait class is based on serial robot dynamics using the Lagrangian formulation. The gait class includes four unique gaits: a forward and a turning gait, which both emphasize speed for the robot; and a forward and turning gait which emphasize traction. Also, we perform an analysis of the variable traction concept.