Unfortunately I don't have practical experience with it but the famous GPars library offers a lot of interesting stuff to handle concurrency problems.
In general I'd say it's as easy in Groovy as it is in Java to mess up concurrency problems. In my experience one of the biggest problems in concurrent applications is mutable shared state and there is not much which prevents the use of mutable shared state, neither in Java nor in Groovy. I think functional languages like Scala or Clojure are better suited for real concurrency problems which doesn't mean that it is not possible to write pefect concurrent applications in Groovy.
I agree that in its core library, Groovy handles concurrency pretty much the same way Java does. The GPars library is extraordinary, however. It has actors, and dataflows, and concurrent collections, and STM, and more. It really deserves more publicity than it's getting.