My idea would be very similar to Part II.

The story would be a contrast between the growing power of the Corleone family under Vito from the 1920s to the 1940s to its ultimate demise in the 1980s and 1990s under Vincent.

Like Part II where the contrast and comparison was between Vito and son Michael, this story would tell how Vito became the powerful and respected man we first see in the original film and how he brought his family to prestige, to Vincent and the steady decline of the empire.

The film would resemble how the times have changed. When Vito came to power during the early to mid part of the century, it was common and relatively easy for the families to involve themselves in many different aspects of making money and acquiring respect and prestige. Fast-foward to the 1980s and 90s (the near end of the century), where the government and the police forces (with new forms of technology) are bringing the mafia to its knees. The times have indeed changed.

Add into this mix the rise of new organizations of crimes (other nationalities, bikers etc) and there is a whole new avenue to tell a story about.

The main focus of the story would be however, is the contrast of the past to the near present. From the glory days of Vito's rise to the grim and dark days of the Corleone family's fall.