Japan Holiday - Spending Advice Needed

Hi
We are going on honeymoon to Japan in September this year. We are visiting Tokyo, Hiroshima, Kyoto and Osaka. My main question is that I keep reading that many places do not accept cards so to carry cash. We are primarily going for the food experience and also to soak up the atmosphere in the different places.
How much cash would we need though for 2 weeks? I know that is difficult to say as different people spend differently but just an idea would be great.
I am aware that as we envisage primarily visiting street food places and smaller restaurants we will definitely need cash for those.
Should we have all the cash we anticipate we need and a bit more exchanged into Yen to take out before we go or should we get enough for the first few days and then get a travel credit card (eg Barclaycard Platinum Travel) to withdraw cash from ATMs after the first few days?
Has anybody been recently who can advise on the current situation? I lot of the websites I'm seeing are a few years old.
We are going in September, so if we need to get a travel credit card I'm probably looking at applying in the next month or so.
How easy is it to find an ATM which will take an international card?
Thanks!