In the small rivers coming from the Coast Range (Luckiamute, Marys River, and Long Tom), it's all about water temperature for the cutthroats. Time of day hasn't mattered much in my experience, but then I'm also not present close to sunrise or sunset.
If the water feels warm enough to take a bath in, then you have to find a deep pool or a tributary bringing in cold water. Shade isn't a factor. Even if the water is reasonably cool to begin with, I notice that the cutthroats still hug the bottom and you just don't see them until they rise up for a surface insect or your spinner.
Over on the Cascade side, the water is more likely to be cool to begin with, but the rainbows and cutthroats both still hug the bottom. I think in that case, it's because the clear waters make them easy prey from the sky.