What cut of beef can I use to make "sukiyaki beef"?
Your best choices would be top sirloin (#1 choice), tenderloin, or one of the other (less expensive) sirloin cuts. Those cuts will be tender, flavorful, and without pockets of fat or gristle to mar the appearance of your dish.

I don't recommend round because I simply don't like its flavor. Using round in this application might be one of the best applications to use up round, but for the best possible beef bowl, I'd go with the sirloin or tenderloin.

TIP: To thinly slice meat, put it in the freezer first until it just starts to get hard (not even close to fully frozen), or start with meat that is only partially defrosted. Partially frozen meat is much easier to neatly and thinly slice. For this application, thin even slices really are important, they make for a much nicer presentation.