The hesitant fuzzy linguistic term set (HFLTS) has turned out to be a powerful and flexible technique in representing decision makers' qualitative assessments in the processes of decision making. The aim of this paper is to develop a method to solve the multicriteria decision making (MCDM) problem within the context of HFLTS in which the criteria conflict with each other. To do so, the concepts of ideal solutions for a HFL-MCDM problem have been introduced. In addition, in order to represent the closeness of one solution to the ideal one, we propose a sort of hesitant fuzzy linguistic measures, such as the hesitant fuzzy linguistic group utility measure, the hesitant fuzzy linguistic individual regret measure, and the hesitant fuzzy linguistic compromise measure. Based on these measures, we develop a hesitant fuzzy linguistic VIKOR (HFL-VIKOR) method, which is motivated by the traditional VIKOR method. The general procedures for the HFL-VIKOR method are given. Some numerical examples are provided to demonstrate the advantages and practicality of our method. Finally, we make some discussions on the advantages of the HFL-VIKOR method, as well as future work.