Greece is a member-state of the European Union and the Schengen Treaty.
Travelling within EU is simply done by showing a national identity. Consequently, the national identity card is sufficient for entering in Greece for citizens who come from EU countries. Nevertheless, for some other transactions (e.g. currency), a valid passport is necessary.
For citizens of the member-states of the Schengen Treaty, a Visa is not required. Contrary, for the citizens of most non-Schengen Treaty countries, a Visa is required to enter both Greece and other EU countries.