Do you need to have your passport on you at all times when traveling in Greece? Can you have a copy of it on you instead?
Kent will come on and give you the thousand postings on this subject. I carry it all the time. It is the only proof I have of who I am and that I am legally in the country. Most of the time you can "probably" get by with a copy but if you really need it one time and you don't have it, what do you do?? A copy is a not substitute for a passport. It is not a big deal to carry it all the time/
I don't know if you need it in Greece (or anywhere else) but I always carry my passport in my money belt. When I'm home I always carry my drivers license on me, not just a copy so I would treat my passport the same way.
If you're constantly on the move from city to city or island to island I'd say yes, MAYBE you should keep it with you. But if you're staying in one place for any length of time and your hotel or your room has a security safe put it away and keep a copy with you. No one is going to ask to see any ID from you except in the most unpredictable of circumstances. Of course the exceptions will be if you want to transact business in a bank, or want to rent a vehicle. Then you need the original along with an International Driving Permit from your local AAA office.
You may never need it, but it is so easy to carry in a neck pouch, why not. I always wear mine and am never aware of it. Be safe and have it with you if you need it. I have needed mine only once while driving in Slovakia and was glad I had it because it made the road stop simple and quick.