This paper analyzes structural waves that propagate freely along an elastic and sagging cable. Results highlight the controlling role played by equilibrium cable curvature, a role not captured in previus studies of cable wave propagation that employ a simple taut string model. The investigation begins with a mathematical model governing the three-dimensional non-linear response of a long elastic cable. An asymptotic form of this model is presented for linear wave propagation along cables having small equilbrium curvature. Upon linearization, structural waves lying within the equilibrium plane decouple from those orthogonal to this plane. The out-of-plane waves are non-dispersive and are well described using the taut string approximation. By contrast, the in-plane waves are dispersive and result in coupled longitudinal/transverse cable deflections. Attention focuses on these coupled in-plane waves for which spectral and dispersion relations are derived. Two qualitatively distinct in-plane wave forms exist. One wave type induces mainly longitudinal deformations and propagates with a speed close to that associated with an elastic rod. The other wave type induces mainly transverse deflections and propagates with a speed close to that associated with a taut string. Free response results demonstrate that significant tension waves may follow from purely transverse initial deflections.