Safety shouldn't be a big concern in Japan, as there is little violent crime against individuals there. Just take your normal precautions: put your money in a wallet and keep your wallet in a safe place, preferably in your pocket (and take care of your pocket!); don't flash your money around; try to keep your bags in sight in crowded places. Having done these, you should be good.
I've once had over 20,000 yen cash right in my pocket in Tokyo and been perfectly safe, but of course I will not recommend this. Don't be like me!
Regarding the cash-centricity of Japan, many small dining establishments (e.g., street-side ramen restaurants & katsudon places) indeed accept only cash and Suica/PASMO. All major convenience stores (7-11, FamilyMart & Lawson), bigger restaurants and virtually all retail places do take credit cards (JCB and Visa, sometimes MasterCard and American Express), although many customers do prefer cash. Thus, if you wish, you don't need to have that much cash on you anyway. Around 10,000 yen of cash should be more than enough for you.