Like others here, I get ready to move left early, usually well before the left turn only lane starts, unless it is a very long lane or there is light traffic. By preparing early, I have more opportunities to find adequate gaps to move all the way over in one safe move. If I get there early, I ride the left side of the leftmost (inside) through lane until the left-turn-only lane starts. This is probably even safer than riding on the right side of the right lane since drivers sit on the right side of their vehicles.
When I reach the LTO lane I ride somewhere between the center of the lane and the right tire track. I don't move out of the way until I have left the intersection on the other side.
-Steve Goodridge