We present a theory of the linear viscoelasticity of dilute solutions of freely draining, inextensible, semiflexible rods. The theory is developed expanding the polymer contour about a rigid rod reference state, in a manner that respects the inextensibility of the chain, and is asymptotically exact in the rodlike limit where the polymer length L is much less than its persistence length L-p. In this limit, the relaxation modulus G(t) exhibits three time regimes: At very early times, less than a time tau(parallel to) proportional to L-8/L-p(5) required for the end-to-end length of a chain to relax significantly after a deformation, the average tension induced in each chain and G(t) both decay as t(-3/4). Over a broad range of intermediate times, tau(parallel to) much less than t much less than tau(parallel to), where tau(perpendicular to) proportional to L-4/L-p is the longest relaxation time for the transverse bending modes, the end-to-end length decays as t(-1/4), while the residual tension required to drive this relaxation and G(t) both decay as t(-5/4). As later times, the stress is dominated by an entropic orientational stress, giving G(t) proportional to e(-t/taurod), where tau(rod) proportional to L-3 is a rotational diffusion time, as for rigid rods. Predictions for G(t) and G*(omega) are in excellent agreement with the results of Brownian dynamics simulations of discretized free draining semiflexible rods for lengths up to L = L-p, and with linear viscoelastic data for dilute solutions of poly-gamma-benzyl-L-glutamate with L similar to L-p.