I've been meaning to post this for a while, now; it's a list of movies, television shows, books, and so on that I think will appeal to Death Note fans based on some element of the story's content. Most of the time, the recommendations I've seen in the fandom extend only as far as Monster and Code Geass.
You can bookmark this entry, or add it to your memories, and I'll update it as I think of more things to list on it.
Movies
Breach: A famous, recent
espionage case presented as a suspenseful drama.
It would be hard to explain this without big spoilers, but you will know why I recommended it by the end of the movie.
The Talented Mr. Ripley: Another murder mystery with a villain protagonist, in which the question is not who the murderer is, but whether or not he'll get away with it.
Visually gorgeous, with strong performances, and seriously slashy. May appeal most to people who ship yaoi pairings in DN, but worth seeing even if you don't.
Real Genius: Super-smart teenagers, at a school modeled on Cal Tech, compete under heavy academic pressure for on-campus positions and post-collegiate jobs... until they learn that someone has been lying to them about the purpose of their lab work.
I see too many fanfic depictions of Wammy's House that make it into something not too far from Kinderheim 511 from Monster: a sinister orphanage in East Germany where the staff and children wind up perpetrating a massacre on each other. That depiction, in which mostly-unhappy and mentally unbalanced residents are pushed too hard in a dark setting, is based mostly on Another Note, which is not by Death Note's original creators. Initial depiction of Wammy's House in the manga showed a place where most of the residents seemed happy to be, in spite of all the intellectual competition; an environment in which they could survive and thrive.
So, I recommend Real Genius as an alternative depiction of a deeply competitive small-scale environment inhabited, for the most part, by young people whose intellectual gifts are profound and whose cleverness is diabolical. Nobody in this film does anything in a minor way; insults and mistreatment are repaid with epic pranks that start big and manage to escalate.
Television
Life: A charismatic and quirky LAPD detective serves a decade in prison for a triple murder he didn't commit. When he is released, he gets a huge financial settlement, and his job is restored to him. He immediately sets about investigating the case in which he was framed.
I loved this show, but fair warning: no one I've recommended it to has liked it! One person, a huge Trek/Whedon fan, thought the pilot was "boring," and another, a criminology major, objected to the main character's unprofessionalism as a police officer. I still recommend it for its strong characterization; I liked the fact that Charlie Crews was off-kilter and interesting. The weekly cases were worth watching, but it was the complex longer plot arcs that made the show satisfying. (I'm dying to write a crossover AU fic in which Crews and Reese work under L, but the prospect intimidates me!)
Criminal Minds: FBI profiling procedural with great characters, for everyone who can't get enough of the NPA Task Force, or all the profiling of Kira that both L and Near engage in.
If you like to see FBI agents following the rules and still getting results, try this one. (At least, I presume they're following the rules, because someone is raked over the coals by Hotchner, their team leader, every time they cross a line.) The Wammy's fans might prefer
NUMB3RS, in which three mathematical geniuses assist the FBI, but its writing is weaker.
The Mentalist: A former celebrity (fake) psychic, who excels at observation and psychological manipulation, is much better at being a detective than the CBI agents he works with, and less bound by Bureau rules. There is one case that haunts him, but in all the others, he mostly just enjoys being the smartest guy in the room.
If you prefer your detectives charismatic and only quasi-professional, this is the one for you. A bonus is the vividity with which the supporting cast is depicted.
Leverage: A team of con artists and criminals with various skills address the balance of power when rich people injure or take advantage of the less fortunate.
Are you a fan of L working as a team with Aiber and Wedy? This is a similar concept, except that their skills are spread between five characters, with the addition of someone with military and combat expertise. Apart from him and the team leader, there's also a hacker, a grifter, and a skilled thief, and a lot of Cunning Plans.