Is there a way to quickly preview spritesheet animation?

Maybe I’m missing something here.
It could be something like selecting 3 grids and then previewing how it looks animated.

No, because Aseprite is not designed to work with spritesheets like this. Your different frames should be frames in Aseprite rather than different parts of the document. That’ll let you preview the animation, control its direction (if you use tags), etc. Then, to get a spritesheet, you can use File > Export Sprite Sheet.