This article is aimed at investigating the plane strain problems in generalized theory of thermoelasticity in a homogeneous, transversely isotropic medium by employing an eigenvalue approach after applying the technique of Laplace and Fourier transforms. An example of a half-space having a step input of temperature has been presented in order to illustrate the application of the approach. The results for coupled thermoelasticity and in the case of homogeneous isotropic medium have also been deduced as particular cases. The results obtained can be used for a broad class of problems in generalized thermoelasticity. The integral transforms have been inverted numerically to obtain the displacement, temperature, and stress fields in the physical domain. The results for these quantities are given and illustrated graphically.