Everywhere we travel (except the Bedouins in Jordan & Egypt) will accept credit cards. Israel, Jordan, and Egypt will accept US dollars but Greece prefers Euros. You will generally get a better price when negotiating if you have cash but you also do not get the protections of a credit card. Sometimes it matters; sometimes it doesn't.
We suggest you have small bills on hand (dollars are ok + some euros) for buying water, ice cream, snacks, and low-cost items. If going to Egypt, you will need many $1 bills for the toilets.