We describe the linear viscoelastic response and dynamics of director orientation in monodomains of unentangled nematic liquid-crystalline polymers (LCPs), using a Rouse-like model appropriate to polymers with a significant degree of backbone flexibility, such as segmented main-chain thermotropic LCPs and side-chain LCPs. We consider the limits of: (i) main-chain directed polymers, for which the polymer backbone is extended along the director, with no hairpins, but executes a random walk in the plane perpendicular to the director, and (ii) anisotropic Gaussian polymers, for which the backbone executes an anisotropic three-dimensional random walk, as for main-chain LCPs with many hairpins and many side-chain LCPs. The analysis is based upon the assumption that, in fluids of such polymers, the distribution of nematogen orientations relaxes rapidly to a state of local equilibrium, via relatively rapid local rearrangements, which follows the slower evolution of the backbone conformation. We describe the linear response in terms of five independent dynamic moduli, which are time- or frequency-dependent generalizations of the Leslie viscosities. We find that directed chains generally tumble in steady shear flow, and that anistropic Gaussian chains flow align. In transient experiments, the coupling between director rotation and the relaxation of backbone conformations is found to lead to an overshoot of the director orientation following a step shear, and a phenomenon of "director recoil" in response to the temporary application of a magnetic aligning field.