If you had to throw a dinner party and invite everyone who had ever stolen the Death Star plans, you'd be surprised at how many place settings you'd have to worry about. -Pablo Hidalgo
Across the various continuities, there are many, many different portrayals. Even accounting for partial plans, many of them completely contradict each other. Some involve Katarn, but most don't.
Star Wars: Dark Forces:
The first mission (which was also released independently as a demo of the game) involved infiltration of an Imperial base on the planet Danuta, in order to steal the Death Star plans. The Mission to Danuta resulted in Rebel possession of the plans that would later be given to Princess Leia and lead to the destruction of the Death Star.
The Death Star Uprising was a relatively large prison riot, originating in Detention Block 2180 aboard the first Death Star, where Rebel Alliance captives-of-war attempted to escape by fighting their way into two hangars and stealing a Sentinel-class landing craft. Although the prisoners' escape was completely prevented, their rioting may have only been a diversion for a more important and ultimately successful objective; a section of the Death Star plans was secretly stolen during the uprising and transported into the possession of Rebels located at a covert base on Polis Massa.
Star Wars: X-Wing:
The Great Search: Rebels discover the Death Star. The campaign introduces Operation Skyhook and ends with the protection of Princess Leia's corvette Tantive IV, which ferries the Death Star plans.
Star Wars: Lethal Alliance:
Rianna escaped the prison on Despayre and managed to make her way onto the Death Star before Despayre's destruction. Upon arrival on the Death Star she headed to the Communication center and told the Rebels about the battlestation. She then stole an Imperial ship and headed to Danuta, where the plans of the Death Star were being held by Kheev. Rianna fought her way to Kheev and killed him in his personal battle armor, though not before Kheev was able to take a pot shot at Rianna. Zeeo sacrificially took the shot and was seriously damaged. Rianna took the Death Star plans from Kheev's body.
The Lost Jedi Adventure Game Book:
Shortly after arriving on the planet, Storm makes a personal enemy of Boba Fett and becomes embroiled in the Rebel Alliance's attempts to steal the plans to a frightening new imperial Superweapon.
Star Wars: Rebel Dawn:
Meanwhile, Bria Tharen led a rebel raid on Toprawa in which rebel spies transmitted the plans for the Death Star I to Leia Organa aboard the Tantive IV. Tharen and her squadron were all killed in the raid, but the plans were successfully transmitted. This was the "first victory" of the Rebels.
Darth Vader arrives on the volcanic moon Kalakar Six in order to intercept Rebels passing on stolen plans for the Death Star.
Star Wars Journal: The Fight for Justice:
The book makes mention of "one Rebel agent [who] managed to get the complete plans" to the first Death Star. It is unclear which agent this is referring to, since no single agent managed to obtain complete plans to the station. Instead, agents such as Kyle Katarn and Havet Storm transferred partial plans to Princess Leia aboard the Tantive IV, where they were compiled into a complete technical readout. Rather than this being a continuity error, it can perhaps be explained by the context in which this information appears: Luke is recounting Leia's story to him of her efforts to deliver the plans to Alderaan. It is possible that Luke either got the information wrong, or that Leia simply summarized the convoluted history of Operation Skyhook.
Tales from the Cantina:
They go to the Dune Sea, and Riij tells them what the Hammertong really is: part of the Death Star II's superlaser. He reveals also he is member of the Rebel Alliance, asks for the weapon, and gives them an offer to join the Alliance. They refuse to give him Hammertong and are ready to return to their home planet, until they learn Riij took their droid with the blueprints of Hammertong inside it. They decide not to go after him and Shada seriously considers joining the Rebels on her own if Karoly refuses.
The Star Wars radio drama:
[See JakeGould's answer.]
The Death Star plans were stolen by Jyn Erso from the Scarif Imperial base, transmitted to a Rebel commandship, transferred to a disk, and sent away on the Tantive IV.
According to the licensed but non-canon Phineas and Ferb: Star Wars, Perry the Rebelpus stole the plans.