Declarative or constraint-based business process and workflow notations, in particular DECLARE and Dynamic Condition Response (DCR) graphs, have received increasing interest in the last decade as possible means of addressing the challenge of supporting at the same time flexibility in execution, adaptability and compliance. However, the definition of concurrent semantics, which is a necessary foundation for asynchronously executing distributed processes, is not obvious for formalisms such as DECLARE and DCR Graphs. This is in stark contrast to the very successful Petri-net–based process languages, which have an inherent notion of concurrency. In this paper, we propose a notion of concurrency for declarative process models, formulated in the context of DCR graphs, and exploiting the so-called “true concurrency” semantics of Labelled Asynchronous Transition Systems. We demonstrate how this semantic underpinning of concurrency in DCR Graphs admits asynchronous execution of declarative workflows both conceptually and by reporting on a prototype implementation of a distributed declarative workflow engine. Both the theoretical development and the implementation is supported by an extended example; moreover, the theoretical development has been verified correct in the Isabelle-HOL interactive theorem prover.