Liquefied Natural Gas (LNG) is natural gas that is converted into its liquid state for logistical benefits. It is also becoming a more viable alternative energy source due to its price competitiveness and environmental friendliness. We study an inventory routing problem for inland distribution of LNG from a storage facility to several filling stations. Here, a transport planner is responsible for the inventory management at the storage facility and filling stations, as well as for the routing and scheduling of a heterogeneous fleet of vehicles. LNG evaporates at a constant rate over time at the storage facility and at the filling stations, and this characteristic relates to the inventory management problems with deterioration described in the literature. Therefore, the problem under study is denoted an LNG Deteriorating Inventory Routing Problem (LNG-DIRP). As a solution method, we propose a matheuristic that combines a mixed integer programming formulation and an adaptive large neighborhood search algorithm. Results of extensive computational experiments confirm the efficiency of the proposed solution method and provide managerial insights to promote LNG as an alternative clean energy solution for the future of transportation. Furthermore, we examine several replenishment policies that are of interest to practitioners and regulatory bodies.