This paper, the first of two related articles, presents a model of viscoelastic deformations associated with earthquakes. A strike slip fault is represented by a rectangular dislocation in a viscoelastic layer (lithosphere) lying over a viscoelastic halp space (asthenosphere). Deformations occur on the three time scales. The initial response is governed by the instantaneous elastic properties of the earth. A slower response is associated with viscoelastic relaxation of the lithosphere, and a yet slower response is due to viscoelastic relaxation of the asthenosphere. The major conceptural contribution of this paper is the inclusion of lithospheric viscoelastic properties into a dislocation model of earthquake-related deformations and stresses. Numerical calculations using typical fault parameters reveal that the postseismic displacements and strains are small compared to the coseismic ones near the fault but become significant farther away. Moreover, the directional sense of the deformations attributable to the elastic response, the lithosphere viscoelastic softening, and the asthenospheric viscoelastic flow may differ and depend on location and model details. The results and theoretical arguments suggest that the stress changes accompanying lithospheric relaxation may also be in a different sense than and be larger than the strain changes.