These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Awesome Music: Beethoven's 'Ode to Joy' from the first film. It's also featured in the trailers for all the sequels.
In terms of music actually written for the movies, Michael Kamen (the first three movies) and Marco Beltrami (who took over following Kamen's untimely demise) turn in some fine work such as Kamen's "The Battle" in the first one and "Shootout And Snowmobile Chase" from the second one, and Beltrami's "Truckzilla" in the fifth.
Cliché Storm: Alik is your typical America-hating Russian villain in A Good Day to Die Hard. The portrayal of Russia in the movie as a whole is a hailstorm of The New Russia clichés that were already dated 20 years ago.
Evil Is Sexy: Maggie Q's character in Live Free or Die Hard, which McClane comments on.
"That girl of yours? Smokin' hot."
Her boyfriend, Thomas Gabriel (played by Timothy Olyphant), is quite easy on the eyes himself.
This trope is pretty much a given when your villain is being played by Alan Rickman.
And then in With A Vengeance, he has to stop a bomber from exploding a bomb on a subway. It would seem Speed is even more like Die Hard (On A Bus and Subway Train) than first supposed.
Genre Turning Point: The first Die Hard practically set the standard by which all future action films were judged, in terms of their heroes, villains, etc. On top of that, it even created an action subgenre.
Harsher in Hindsight: In Live Free or Die Hard, the Big Bad's apparent motivation for messing with America's computer infrastructure is to teach them a lesson for ignoring him about the threats cyberterrorist can pose. In Real Life, Dr. Bruce Edward Ivins was suspected to have caused the 2001 Anthrax attacks for this same reason.
In With A Vengeance one of the cops makes an offhand comment to the 1993 bombing of the World Trade Center, and how darned inconvenient it was. Ouch. The shot of McClane and Zeus running through New York with the towers clearly in the background doesn't help.
At one point John says there's a bomb in the trash can and everyone around him thinks he's nuts. If that happened today the whole street would be in a panic.
Looking at the first three films: a tower gets blown up, a terrorist plot with airplanes, and New York City as the target of terrorist attacks.
A related, but non-Japan example: A Good Day to Die Hard was a critical success in Indonesia.
Jerkass Woobie: Karl in the first film. He shows real grief and rage when his brother Tony becomes the first baddie McClane kills, and loses all interest in the robbery for the rest of the film, so bent is he on avenging his sibling, and who wouldn't be furious if their sibling was killed? (True, McClane only killed Tony in self-defense, but still.)
Magnificent Bastard: Hans Gruber, especially when he gets caught at gunpoint by McClane and almost gets away with simply using an American accent to pretend to be a civilian. And even then, after his henchmen turn up, he manages to do some serious damage to McClane by taking advantage of his bare feet and shooting the glass. Also, his brother Simon. It runs in the family.
Sequelitis: A Good Day To Die Hard is, at least from a critical standpoint, the weakest movie in the franchise (given the largely negative response, far and away below the other four). While the first four films had enough plot to fill two hours, the fifth barely has enough material to scratch 90 minutes, a bulk of that being action and the rest being exhausted The New Russia cliches.
Strawman Has a Point: Gabriel did try to warn the government who hired him that their security was woefully inadequate...It still doesn't stop him from exploiting those weaknesses to get rich and kill people (and not in that particular order).
Weird Al Effect: Everybody knows McClane's catchphrase "Yippie-Ki-Yay, motherfucker!", but few now remember that the line (or at least the "Yippie-Ki-Yay" part) was derived from the western song "Git Along, Little Dogies".