We present a study demonstrating the capability for controlled shape memory thin-film growth using molecular beam epitaxy for the NiTiCu alloy system. The martensitic transition in these films occurs at film thicknesses well below 1 pin with comparable transition temperatures as known for thick sputter-deposited films (1-10 mu m). Mechanical stress measurements reveal a stress recovery on the order of 300 MPa. Single- or two-stage transformation is observed depending on the quantity of the Cu additions which is discussed in the context of a load-temperature phase transition diagram. As for the microstructure, the crystallites are highly oriented along the film plane normal. Moreover, splitting of the martensite orientation is detected, indicating the selection of two sets of martensite variants with defined orientation characteristics. The occurrence of this preferential order is deduced to the formation of a Ti-rich (Ti2Ni) phase at the film-substrate interface providing a highly oriented initial crystallization template. In this respect, the relevance of Ti2Ni precipitates is addressed. The results described in this study outline the perspectives for tailoring the microstructure and transformation performance of shape memory thin films, especially in regard to technological applications.