The Mariners can't wait for the slackers to "wake up"... so let's just bat the producers at the top!
Start the lineup with Ichiro, Griffey, then Branyan. Follow with whoever else.
They're all lefties, which is ultimately to Branyan's advantage:
Ichiro is top five in the AL in batting average, and is a threat to steal. He therefore sucks up a lot of the pitcher's mental energy: first, they have to pitch him gingerly to keep him from getting on, and then they have to divide their attention to keep him from stealing.
Griffey is a historical power threat, and is also willing to take a walk. He therefore sucks up a lot of the pitcher's mental energy too, but for different reasons. The pitcher is not afraid of Griffey getting the dribblers, bloopers, infield singles, and occasional liners that Ichiro does, but he is definitely afraid of what's left of Griffey's power. So the pitcher will pitch quite gingerly to Griffey as well, and in the end, after all that effort, Griffey might draw a walk.
Then comes Branyan, who has been watching this whole thing transpire. By watching Ichiro, Branyan learns how they're attacking a singles hitter. By watching Griffey, Branyan learns how they're attacking a power hitter. By the time Branyan actually steps up to the plate, it's almost as if he's had two practice at-bats already. He knows whether the pitcher is prone to small ball, or prone to long ball. Or maybe neither, which means Branyan should try a different approach altogether. In every case, seeing how the pitcher handles two lefties on opposite ends of the hitting style spectrum provides valuable information to Branyan.
As for the rest of the lineup... I don't know. Just tell everybody to try to draw a walk, to try to get rid of the starting pitcher before the 6th inning.