The LLY -- 04.5--05 years had a connection problem with the injectors, and a wear through on the FICM harness-- both an easy fix. Less common is some higher mileage trucks lost the head gaskets.
The 06-- can be a LLY or a LBZ--same engine and all parts-- the LLY had a mild tune. No real problems-- probably the best Duramax
Chevy 04.5 LLY, 2X, shell, XDR, MikeL Heavy duty up-grade, 6spd, TransGo, deep pan, de-badged, 65 Gallon replacement Aero tank, LBZ-mp, Kennedy pump, shimmed, Pre-fuel filter, tow mirrors, BU cam, gauges, GPS, Sirius, bags, custom grille, fender vents, HD trans cooler- Curt front hitch, footwell lights, load lights, Studs, head gaskets.