This paper presents a Large Neighborhood Search (LNS) approach based on constraint programming to solve cumulative scheduling problems. It extends earlier work on constraint-based randomized LNS for disjunctive scheduling. A breakthrough development in generalizing that approach toward cumulative scheduling lies in the presented way of calculating a partial-order schedule from a fixed start time schedule. The approach is applied and tested on the Cumulative Job Shop Scheduling Problem (CJSSP). An empirical performance analysis is performed using a well-known set of benchmark instances. The described approach obtains the best known performance reported to date on the CJSSP. It not only finds better solutions than ever reported before for 33 out of 36 open instances, it also proves to be very robust on the complete set of test instances. Furthermore, among these 36 open instances, one is now closed. As the approach is generic, it can be applied to other types of scheduling problems, for example problems including resource types like reservoirs and state resources, and objectives like earliness/tardiness costs and resource allocation costs.