Believe it or not they're still loading screens. They just "thread" the loading so it's not as apparent.
Threading is a way to 'do two things at once in a single program'. In this case, "Thread 1" is the gameplay program, and "Thread 2" is the loading of sprites, level data, etc. While you're in the transition room, Thread 1 plays the game, you jump/moving through the room etc, and Thread 2 loads the data for the next segment during that. This way gameplay is not interrupted in the process.
Not all consoles have this capability for the core game code. It's something that's fairly new.
Here's an extra trick you can do to reveal the hidden "load time" in the DS titles. Strap on as much speed increasing equipment and buffs as you can, and speed through the Transition Room at super speeds. When you reach the other exit of the room, the game will pause longer than usual before you're in the next room. This is because Thread 2 was not finished loading everything yet, and forces Thread 1 to pause and wait until the next room is finished setting up for you to enter.
Another tidbit of information is that the effect that threading gives you can be observed on older consoles too. Sound in a console game is handled by a different processor than what the game's main code runs on. This is why when a game has slowdown, the music and sound effects also don't exhibit any slowdown or distortion. The processor the game is running on may be slowing down due to whats happening in the game, but the music chip is it's own unit and will continue processing at it's normal speed regardless of what's going on in the game. This is indeed different than the game itself being able to run two threads at once though. Threading leads to multiple processes on the same chip, as opposed to different chips running their own single process.