Keith,
Any traditional Greek restaurant/taverna will have "vegetarian" food - i.e traditional dishes that do not include meat. Try gigantes (giant baked beans in tomato & oil), imam (stuffed aubergine), briam (like a layered vegetable stew) and the runner beans in tomato & oil (can't remember the name of the dish, something like fassolia). Even the touristy restaurants (and there are quite a number in Pefkos) will do Greek salads and some do vegetable mousaka.
What will be difficult/impossible to find are vegetarian versions of "western" dishes (veggie sausages, veggie burgers etc) but the Greek food is so good you would not want/need these anyway!

One good way of finding your feet with Greek food is to order a meze (there are "meat" and "non-meat" versions) where you get lots of small plates of various dishes - great for sharing.

Cheers,
Jet

If you visit Lindos one evening i can highly recommend Timis Cyprus Taverna. They do an amazing vegetarian meze for 2, just don't eat anything the rest of the day or you'll never get through it!

Timis is opposite Lindian Apollo Bar, just by the steps that lead up to The Acropolis.