Depending upon the cuts and dishes, different cutting methods are used for. Here's an overview of the different techniques for Japanese Wagyu beef.
Cut crosswise into strips (at right angle).
This cutting helps to maintain tenderness of meats and heat goes faster into center of meats.
Typically used for marbling meats like ribeye and striploins.
Cut along stripes in parallel.
With this cutting, heat goes more slowly into center of meats.
Thus, this is good for any dishes using oils.
Typically used for lean meats like rounds.
Cut on an angle against strips.
This cutting is middle of steak cutting and katsu cutting.
With this cutting, you can get wider surface cross-section, so it is typically used for tender cuts for BBQ (Japanese or Korean style).
Important point of this cutting is to keep all in the same size.
This cutting is typically used for simmered dishes like stew, curry etc. or even great for one bite steaks.