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 Lp. In this limit, the relaxation modulus G(t) exhibits three time regimes: At very early times, less than a time τ∥∝ L8Lp5required 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, τ∥≪ t ≪ τ⊥, where τperp;∝ L4/Lpis 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) ∝ e-t/τrod, where τrod∝ L3is a rotational diffusion time, as for rigid rods. Predictions for G(t) and G* (ω) are in excellent agreement with the results of Brownian dynamics simulations of discretized free draining semiflexible rods for lengths up to L = Lp, and with linear viscoelastic data for dilute solutions of poly-γ-benzyl-L-glutamate with L ∼ Lp.