The essential mechanism underlying animal locomotion can be viewed as mechanical rectification that converts periodic body movements to thrust force through interactions with the environment. This paper defines a general class of mechanical rectifiers as multi-body systems equipped with such thrust generation mechanisms. A simple model is developed from the Euler-Lagrange equation by assuming small body oscillations around a given nominal posture. The model reveals that the rectifying dynamics can be captured by a bilinear, but not linear, term of body shape variables. An optimal gait problem is formulated for the bilinear rectifier model as a minimization of a quadratic cost function over the set of periodic functions subject to a constraint on the average locomotion velocity. We prove that a globally optimal solution is given by a harmonic gait that can be found by generalized eigenvalue computation with a line search over cycle frequencies. We provide case studies of a chain of links for which snake-like undulations and jellyfish-like flapping gaits are found to be optimal.