Have a think about Lefkas:
http://www.lefkas.net/
I went there last year and stayed about half an hour’s walk from Lefkas town. Although they cater for tourists there are no noisy bars, certainly no Karaoke bars or football bars. Most of the tavernas are family run (and they take you into the kitchen to choose your food instead of having a menu). The bars we used were quiet with occasional visits from local musicians (plinka-plonk-plunk stuff, bouzoukis and all that). You get Brits staying there but (and dare I say it) of the less yobbish type, usually families or couples. It's very much a Greek town that has foreign guests staying round and about rather than a "resort" which has been overrun by the elements you (and I) are obviously not too keen on.
We found it a quiet but lively town. We had a nice villa with a pool so did not travel about much, but we did come across one or two nice beaches with just a bar and/or a taverna for sustenance. We found it very enjoyable.