The most hygienic restrooms are found in hotels and department stores, and are usually clearly marked with international symbols. You may encounter Japanese-style toilets, with bowls recessed into the floor, over which you squat facing the top. This may take some getting used to, but it's completely sanitary as you don't come into direct contact with the facility. If you can't face a squat, check out the last cubical in the row because it may be a Western-style toilet.
In many homes and Japanese-style public places, there will be a pair of slippers at the entrance to the restroom. Change into these before entering the room, and change back when you exit.
Many public toilets don't have toilet paper, though there are dispensers where packets can be purchased for ¥50 or so. Many locals accept the free tissue packets that are handed out as advertisements in the centre of town for this reason. Similarly, paper towel dispensers and hand dryers are not always installed, so bring a small handkerchief or washcloth with you, as well as some hand sanitizer.