We perform Monte Carlo analyses of the anisotropic viscoelastic response of random RVEs representing the microstructure of UD fibre composites. Both fibres and matrix are taken as isotropic viscoelastic solids; the fibres have different cross-sections including circular, elliptical, Reauleaux and star-shaped; they are separated from the matrix by interphase regions of different mechanical properties and thicknesses. The analyses allow determining the sensitivity of the transversely isotropic, viscoelastic response of UD composites to fibre volume fraction, fibre shape, interphase volume fraction and interphase properties.