One option is to screw cleats perpendicular to the joists.

Benefits:

• load of the cabinets + counter will be evenly distributed on a 2x4 which is perpendicular to 3 joists.

Downsides:

• I will have to screw the base cabinets through the sides of the island. This way, screws will be visible. When the cleat is parallel to the joists, my kitchen layout will allow me to screw through the toe kick on the cabinets, making the screws less obvious.
• Load is still mostly distributed across 3 joists. Is that okay?

I'm not really sure if this is the way to go, any comments?

One option is to screw cleats perpendicular to the joists.

Benefits:

• load of the cabinets + counter will be more evenly distributed on 2x4's which are perpendicular to joists.

Downsides:

• I will have to screw the base cabinets through the sides of the island. This way, screws will be visible. When the cleat is parallel to the joists, my kitchen layout will allow me to screw through the toe kick on the cabinets, making the screws less obvious.

I'm not really sure if this is the way to go, any comments/suggestions?