I was pleasantly surprised how widely accepted it was in Japan, you shouldn't have any problems in major shops, department stores or restaurants. Smaller businesses are hit and miss with cards general, normally I would just carry cash for any purchases of less than 10K Yen. From what I understand though, American Express have a deal with JCB, so any retailer that accepts JCB cards (by far the most used card in Japan) has the option to accept Amex too, whether they do is their choice though.
My American Express card is actually fast turning into my card of choice when I travel abroad. Unlike my visa, they never block it, they don't charge me a fee (unless I'm getting cash on it) and the conversion rates actually seem to be quite generous, rather than screwing me over.