Well - After a couple of years residing in Japan I decided to get a United Airlines mileage credit card. Was refused by Visa, Master (twice I think).
They would not tell me why - told me to contact the company that rates credit-worthiness. I did so - discovered that there were no issues with my credit rating.
Finally I contacted JCB - who sent me the application forms with translations into English (nice touch). I completed the forms and they issued a credit card which earns me miles with United. Nice.
The draw backs were that the card is not widely accepted outside Japan - and the credit limit was incredibly low at the start. I had to maintain my UK credit cards (AMEX and visa) in order to have access to enough credit when travelling.
I persuaded them to increase my credit limit - so these days it meets most of my needs and keeps adding miles to my UA account.

People have suggested to me that I try VISA JAPAN and MASTERCARD JAPAN again - since I would have less trouble using their cards in the US and UK. However, those companies left such a bad impression with me that I don't feel like doing business with them.