A semi-analytical method for the quick and robust solution of guided waves excited in infinite laminated composite strips is presented. In the proposed method, the response of the strip is constructed in time-space domain directly from the governing wave equations, assuming a layerwise variation of the through-the-thickness displacement components. Analytical Fourier transforms, with respect to time and space, provide the transformed response in the frequency-wavenumber domain. Dispersive modal properties for the structure are extracted. Finally, the time transient response of the propagative guided waves is calculated using two inverse Fourier transforms, the first is calculated via the Cauchy's residue theorem, whereas the second is numerically computed. Various numerical results are presented for isotropic, cross-ply composite laminate and sandwich strips and are compared with results from other reported methods as well as with experimental results in composite strips.