The giant magnetoresistance of crystalline Fe/Cu/Fe(001) epitaxial structures characterized by scanning tunneling microscopy are presented. Fe/Cu/Fe(001) trilayers capped with Au are grown directly on GaAs(001) using a new procedure for producing pure Fe layers with As-free Fe surfaces on GaAs(001). The temperature dependence of the magnetoconductance and sheet resistance measured from 4 to 300 K is modeled by the Boltzmann equation assuming that the mean free paths in the crystalline epitaxial layers are equal to those in bulk materials. The results of the simple model suggest that the coefficient of the specular scattering at the Fe/GaAs interface is R=0.45, while the scattering at the outer Au interface is diffuse. Spin asymmetry scattering at the metallic interfaces is Delta T=T-↑-T-↓=0.34, T-avg=(T-↑+T-↓)/2=0.79. The sheet resistance was best modeled using a low temperature mean free path of 25 nm in the Fe layer.