DAG has been extensively used in Grid workflow modeling. Since Grid resources tend to be heterogeneous and dynamic, efficient and dependable workflow job scheduling becomes essential. It poses great challenges to achieve minimum job accomplishing time and high resource utilization efficiency, while providing fault tolerance. Based on list scheduling and group scheduling, in this paper, we propose a novel scheduling heuristic called DAGMap. DAGMap consists of two phases, namely Static Mapping and Dependable Execution. Four salient features of DAGMap are:(1) Task grouping is based on dependency relationships and task upward priority;(2) Critical tasks are scheduled first; (3) Min-Min and Max-Min selective scheduling are used for independent tasks; and (4) Checkpoint server with cooperative checkpointing is designed for dependable execution. The experimental results show that DAGMap can achieve better performance than other previous algorithms in terms of speedup, efficiency, and dependability.