In this paper, we analyze the global convergence of a general non-monotone line search method on Riemannian manifolds. For this end, we introduce some properties for the tangent search directions that guarantee the convergence, to a stationary point, of this family of optimization methods under appropriate assumptions. A modified version of the non-monotone line search of Zhang and Hager is the chosen globalization strategy to determine the step-size at each iteration. In addition, we develop a new globally convergent Riemannian conjugate gradient method that satisfies the direction assumptions introduced in this work. Finally, some numerical experiments are performed in order to demonstrate the effectiveness of the new procedure.