Let X = G K be a Riemannian symmetric space of the noncompact type and let L_X be the Laplace-Beltrami operator on X. We consider on X × R a differential equation of the type L_Xu = P(∂_t)u, where P(∂_t) is a second order differential operator in t ε{lunate} R. Using the Radon transform on X we relate Huygens' principle for the solutions u to this equation, to the corresponding question for the solutions v to the equation L_Av = (P(∂_t) + R)v on a maximal flat subspace A, where R is a certain explicit constant. In particular we conclude that Huygens' principle holds for solutions to the (modified) wave equation on X obtained by letting P(∂_t) = ∂_t² - R, when X is odd-dimensional and G has one conjugacy class of Cartan subgroups.