Greengrass is no stranger to Iraq. Ten years ago he made a movie for British television called The One That Got Away, about an operation during Desert Storm. His hand-held documentary style is perfectly suited for the discord of the region and its post-invasion events, though he will probably bring something original to the project that hasn't yet been seen in the actual documentaries coming out of the Iraq War.
Despite his putting out mostly apolitical work since arriving in Hollywood, he is typically a political filmmaker and we can probably expect a very critical position from him here. Chandrasekaran's book apparently reveals the failures of the U.S. occupation and presents the whole thing as a major historical disaster. The Washington Post journalist does depict some involved individuals in a good light, though. If Greengrass can keep it as honest and balanced, he might be able to keep some of his patriotic audience.