Walk slowly and carefully, looking both close and further into the bush. Stop for a couple of seconds and listen, and look with your eyes, trying to detect the slightest movement. Walk 10-20 paces, stop again. Look behind you also, sometimes they'll break cover after you've gone by. ALWAYS have the gun at the ready position; the moment you've got it over your shoulder, or in one hand, or while you're taking a leak is when they'll flush. Don't be afraid of walking down a game trail, and circling back through the bush to your main trail. If you see a track in the snow, follow it along.
If you hear a flush but don't see the bird, don't panic. Listen if you can for which direction it went. Then carefully look around for a second bird which DIDN'T flush from that spot. Then follow that first flush. Remember, that bird will fly to a spot, and then run off either left or right to try to put you off their direction. So go slowly, look and listen, especially to the left and right. You can often reflush that same bird, so keep after it.
If you shoot, the bird will generally go down immediately if it's hit at all. They're not like geese, which can sail off if wounded, nor sharptails who can take a hit, bounce, and then take off running at full speed. Ruffies will drop and flutter if wounded.
Finding grouse territory is the most important thing. Mixed trees, low cover, open areas, low or little grass, some thick shrubs or windfalls, are things to look for.
Because I'm right-handed, my gun barrel points to the left. So I'm ready for a flush to the left almost instantly. So I'll focus my searching a little bit more on the right side as it will take a split second longer to bring the gun around, knowing that if I hear a flush on the left I only have to turn my head. If I'm walking along a transition between bush and open area, I'll keep the bush on the left.
Best times are sunrise + 2 hours, and 2 hours before sunset/last light when they're out feeding.
A gun with no choke, really, is best for wingshooting these little feathered ninjas ( I like that term! ). You want as many pellets in the air as possible, so 7-1/2 or 8 shot is preferred. Remember that 20 and 28 gauge guns are very effective, so you don't need 1-1/2 ounce loads of buffered copper plated #5. The last two birds I shot, one going straightaway, the other quartering away, neither had a single pellet in the breast. A leg was broken, wings were broken, head was bleeding, but the body stopped that light shot before it got into the meat. Not always perfect like that.
Hope you get some birds!