What are some methods to deal with cleanliness when working with raw chicken, pork, etc.?
Commercial kitchens use a few methods that are hidden from the camera during kitchen shows. First, there are cutting boards for meats, and separate cutting boards for vegetables. These are often color coded so they aren't mixed up. If there is room, raw meats use an entirely different table. Any possibly contaminated board is sanitized before reuse.
Second, the cloths you see are stored in a bucket of sanitizing fluid which kills microbes on contact. When a surface is wiped down, or hands are cleaned, they are being sanitized by the liquid on the cloth. Finally, there is usually more than one person working behind the scenes and labor is divided so the raw meat people are not the same as the green salad people.

At home, planning is the key. Try to do all of your non-meat prep first. Then you can contaminate your cutting board without having to wash it again. Then try to take care of all of your meat prep, so you only have to wash your hands once. And remember that if you drop a vegetable on your raw chicken, it's not a big deal if it's all going to be cooked.