Ninja Run

Related to Airplane Arms. The standard Ninja Run is this:

Body tilted forward, low to the ground.
One forearm crooked in front of the body, apparently to block attacks (two fingers held in front of the mouth optional but prevalent).
The other arm behind the back, ready to pull out a bladed weapon.

If the ninja carries a blade while running like this, it is always held in a Reverse Grip.

Compare with Airplane Arms, where both arms are held straight out or swept backwards.

Examples of Ninja Run include:

Anime and Manga

Renamon from Digimon Tamers
Black Star and Tsubaki do this, along with the jumping through trees thing.
The monster Strike Ninja from Yu-Gi-Oh!. He even holds his dagger in a reverse grip with the front arm. As Joey puts it, "I love watching this guy run."

Fan Works

Mel C gets called out for naruto running during a chase in Final Stand of Death.

Literature

Jiro and most of the other ninja from Dagger of Kamui. One scene has Jiro and his freshly hired Redshirt Army doing a Power Ninja Run against a rising sun, even drawing their blades in unison.

Live Action TV

In the third season of Mighty Morphin Power Rangers, the Shogunzords ran like this before combining into the Shogun Megazord. Unfortunately, since the Shogunzords were pretty bulky and blocky, it looked less like a Ninja Run and more like they were running and checking their watches at the same time.

MMORPG

This was added as a travel power as part of the purchasable "Martial Arts Super Booster Pack" for City of Heroes. It made an excellent substitute travel power for character concepts that wouldn't exactly be able to leap tall buildings or run at the speed of light. It was even called "Ninja Run". Since the game's resurrection in 2019, it's available for free from special vendors in the tutorial and starting zones.
Shaiya Assassins and Rangers are fond of doing this, as they're very much ninja-like in this game.
In Lineage 2, the Dark Elves running animation(but females use without the forearm crooked in front of the body).

Video Games

The demon hunters in Warcraft III.
Strider, but only in the Capcom vs. Whatever games and the second Strider game.
The player character in Jade Empire runs like this when in focus mode. Also, whenever Master Li runs.
Mai Shiranui runs exactly in this way in the King of Fighters series, despite not carrying any weapon on her back.
Phantom from Mega Man Zero.
Yoshimitsu from Tekken can be seen doing this in his victory cinematic from Tekken 5.
Ryu Hayabusa, (duh), does this in cutscenes. In game, he runs like a normal person. Or maybe more like a normal person who was awesome.
Shinobu in No More Heroes.
Ninja chicks (Midori, Mizuki, Luna, Misa) in the arcade The Punisher. And they do this in high heels.
Joe Musashi of Shinobi.
Taki from the Soul Calibur series.
Espio the Chameleon in Sonic the Hedgehog.
Sheik in Super Smash Bros..
Guy in the Street Fighter series.
Shen in League of Legends moves in this manner, complete with the index and middle finger of the front arm outstreched and the other arm behind his back, ready to throw blades. His incredibly slow pace really makes it more of a Ninja Stalk, however.
Kotaro in Sengoku Basara runs like this, with one hand on his sword hilt. Sasuke and Kasuga too, although both holding a weapon in each hand. Also Magoichi, despite not being a ninja.
Assassins in Guild Wars do this while under the influence of a speed boost. To their credit, unless they're wielding daggers at the time they will sprint in a more standard fashion; daggers are always dual-wielded and held in a standard grip in both hands, meaning it's actually fairly practical to run like this (one dagger is ready to block attacks, the other one is in a position to use as much of your momentum as possible for a strike).
Sol Badguy, Milia Rage and Chip Zanuff in Guilty Gear.
In The Elder Scrolls V: Skyrim, the Player character does this when sprinting in Third-Person while Dual-Wielding.

Western Animation

In the DC Animated Universe, Batman often runs in a similar manner with both arms concealed beneath his cloak, presumably to quickly draw any weapons or tools and without telegraphing his actions.

Real Life

A modified version of this run is used by modern soldiers to reduce their profile. The soldier runs partially bent or hunched over with head down so that any bullets aimed at his head will strike the helmet, and with his weapon held closely in front of him.