An assessment of five different definitions of the principal molecular axis along which molecules align in a nematic liquid crystal system has been made by analysing fully atomistic molecular dynamics (MD) simulations of a set of anthraquinone dyes in the cyanobiphenyl-based nematic host mixture E7. Principal molecular axes of the dyes defined by minimum moment of inertia, minimum circumference, minimum area, maximum aspect ratio, and surface tensor models were tested, and the surface tensor model was found to give the best description. Analyses of MD simulations of E7 alone showed that the surface tensor model also gave a good description of the principal molecular axes of the host molecules, suggesting that this model may be applicable more generally. Calculated dichroic order parameters of the guest-host systems were obtained by combining the surface tensor analysis with fixed transition dipole moment (TDM) orientations from time-dependent density functional theory (TD-DFT) calculations on optimised structures of the dyes, and the trend between the dyes generally matched the trend in the experimental values. Additional analyses of the guest-host simulations identified the range of conformers explored by the flexible chromophores within the dyes, and TD-DFT calculations on corresponding model structures showed that this flexibility has a significant effect on the TDM orientations within the molecular frames. Calculated dichroic order parameters that included the effects of this flexibility gave a significantly improved match with the experimental values for the more flexible dyes. Overall, the surface tensor model has been shown to provide a rationale for the experimental alignment trends that is based on molecular shape, and molecular flexibility within the chromophores has been shown to be significant for the guest-host systems: The computational approaches reported here may be used as a general aid in the predictive design of dyes with appropriate molecular shapes and flexibilities for guest-host applications.