I wanted to play L.A Noire at one point but watching streams made me decide not to, don't remember exactly why. But as for something similar if you ever find Mafia 1 and 2, not sure if 3 is released for 360, you can give them a try, these should be pretty cheap to buy if found.

I replayed them last year, the remastered version for PC, had so much fun. The games being pretty old have simple mechanics but for their time I'd say they were a little ahead.