I saw this in my teens, and I wasn't all that impressed. I kind of forgot about it for a while, and then I started hearing that it was some kind of unsung classic of the eighties. I decided to give it another try.
A 1950s private detective named Harry Angel is hired by a guy named Louis Cyphre to find a missing person. In case you don't get the joke right off the bat, just keep repeating the name Louis Cyphre over and over until you do get it. Anyway, the guy Cyphre is looking for is a singer who was supposedly knocked into a coma during World War II, and so he was never able to pay off on a deal he made with our friend Lou. Now, Lou thinks that this singer isn't really in a coma, and he wants Harry to find him. As soon as Lou mentions the war, Angel brings up that he was in the war and had been "fucked up" in some undisclosed way. Okay, so who's figured out the ending of the movie? Even though it's obvious where all this is leading, there's still an investigation to sit through. Harry wanders around New York and then heads down to New Orleans to ask questions to people who turn up dead later. He also keeps talking about how he has a "thing about chickens." I don't know what the hell's up with that.
Angel heart is dull, predictable, and a little on the dumb side. There is some nudity and a little bit of gore, so it's not a total loss.