EU nationals don't need a work permit, but they need a residency permit and a Greek tax-file number if they intend to stay longer than three months. Nationals of other countries require a work permit.
Bar & Hostel Work
The bars of the Greek islands could not survive without foreign workers and there are thousands of summer jobs up for grabs every year. The pay is not fantastic, but you get to spend a summer on the islands. April and May are the times to go looking. Hostels and travellers' hotels are other places that regularly employ foreign workers.
If you're looking for a permanent job, the most widely available option is to teach English. A TEFL (Teaching English as a Foreign Language) certificate or a university degree is an advantage but not essential. In the UK, look through the Times educational supplement or Tuesday's edition of the Guardian newspaper for opportunities; in other countries, contact the Greek embassy.
Another possibility is to find a job teaching English once you are in Greece. You will see language schools everywhere. Strictly speaking, you need a licence to teach in these schools, but many will employ teachers without one. The best time to look around for such a job is late summer.
The noticeboard at the Compendium bookshop (www.compendium.gr) in Athens sometimes has advertisements looking for private English lessons.