Civil Rights History Project Act of 2006 - Requires the Librarian of Congress and the Secretary of the Smithsonian Institution (acting through the Director of the National Museum of African American History and Culture) to establish an oral history project to: (1) collect video and audio recordings of, and visual and written materials relevant to the personal histories of, participants in the Civil Rights movement; and (2) make the collection available for public use through the Library of Congress and the Museum.