The role of the acyclicity property of fuzzy preference relations is studied in the framework of rationality of fuzzy choice functions. Two standard ways of constructing a fuzzy choice function from a given fuzzy preference relation are considered and properties such as acyclicity and completeness are shown to be sufficient to ensure the rationality of the fuzzy choice function. Special attention is paid to the triangular norm used for modelling the conjunction. The results obtained are compared to the classical results on rationality of crisp choice functions. Finally, the well-known Richter theorem is investigated in the fuzzy setting.