In my opinion, it is good to travel with at least a small amount of euros to get you through the airport, a quick coffee, and the transfer to your destination. If you don’t already have some euros, AAA always has $200 packs available. Of course these come with their fee and not a great exchange rate, but it’ll get you started.
Once there, you can readily access euro cash from a bank-affiliated (NOT private, tourist) ATM, using your no-fee card. All banks in Greece now charge a small fee, but generally the National Bank of Greece has the best rate. I usually carry a debit card, because my credit card goes through ATMs as a cash advance, with heftier fees and a ceiling limit.
Over the years it is true that more places in Greece are accepting credit cards, but whenever possible I still personally prefer to use cash, except for major purchases like hotel, car rental etc. However if you do use your card, ask for it to be rung up in euros - not dollars, if given the choice.
This year is going to be a little different as Greece re-opens to tourism. Again this is strictly my personal decision, but I am going to be bringing more cash with me in order to be prepared for any unforeseen problem. This is a terribly long reply, but I hope it helps.