The stability of plane steady oscillations of a slender elastic rod subjected to lateral harmonic excitation is investigated. For certain values of excitation amplitudes and frequencies the planar response is unstable, and nonplanar motions are parametrically excited. Steady nonplanar motions are characterized by each point on the rod centerline tracing an elliptical path perpendicular to the rod axis. The investigation is restricted to cantilevered rods with nearly equal principal moments of inertia of the cross‐section, and excitation is always in the direction of one of the principal axes of inertia. The effects of damping and of slight variations of the natural frequencies associated with motions in the two principal planes are investigated. Supporting experimental evidence is presented. Finally, the principal features of the transversely driven rod response are compared with the response for the same type rod excited axially.