Anthrax Attacks Investigation Act of 2008 - Establishes the National Commission on the Anthrax Attacks Upon the United States to: (1) conduct an investigation of the anthrax letter attacks of September and October 2001; and (2) submit interim reports to the President and Congress and a final report on its findings with recommendations for corrective measures to prevent and respond to acts of bioterrorism. Directs the Commission to hold public hearings and release its reports to the public.