I would argue that the major way in which our historic roots have affected modern criminal investigations comes in the way that the 4th, and to some extent 5th, amendments have influenced our system.
These amendments were created because of the Founders' fear of intrusive government. This fear of government is one of our major historical roots. The amendments that came out of it have made it so that criminal investigations even today must be conducted with a great deal of sensitivity to the rights of the accused. We protect the rights of the accused against unreasonable searches and against forced confessions. We also protect (6th amendment) their right to be tried by a jury.
All of these protections affect the way that criminal investigations are carried out. They cause the American criminal system to focus as much on rights as on preventing crime and maintaining order. This is a clear legacy of our founders' fear of government.