I spent a few weeks cycling in Taiwan recently. My wife and I brought our own bicycles from the 'states and rode part of the way around the island, mostly the scenic east coast, and as well we rode inside Taipei briefly.
"Bike Boxes" in Taipei (and other larger cities in Taiwan) are really motor-scooter boxes. Effectively, the box takes up one or two right lanes and precedes the crosswalk lines about 10 feet. There's about 200 square feet of space. At red lights, the motorbike (and cyclist) traffic is supposed to filter forward and take this space.
Unless a car were to turn (illegally) right from the left of the box, you're effectively out of danger of being hit from a right hook. You're also much more visible to drivers and you're able to take your "free" right before other motor traffic. I'm not really that keen on bicycle lanes for safety, but I found this a dramatic improvement of safety, with not so very much inconvenience to cars.
The areas were denoted with white lines and a figure of a helmeted motorcyclist. Drivers kept out of them. Some areas, like Taitung, had separated lanes for motorbike and cars.