Is it possible, wise to rent a furnished place for a 3 month stay?

Your idea is brilliant and it is going to save you a lot of money but It is not as easy as it sounds. First you have to find the place and then get it at good price too. The earlier you book it the better will be for you and your pocket.
Three months is long enough, I am pretty sure you'll be able to find something.
can i just ask which island are you thinking of going? I am asking this because the popular islands do not offer that kind of accommodation. You might be able to find a studio flat etc but the rates will still be pretty high.

Well, seeing as my knowledge of the country is slim, I had by default figured on going to one of the ones I see named most often in the popular photographs--Santorini, for example.  But I don't quite know what the best way to "shop" for a place is, from halfway around the world.  So far this forum has been helpful with getting some idea what costs are, and how some of the transportation options work.  I have also learned to bring my own books (in English) and as much of my personal toiletry items as possible (like makeup and creams).  Though I don't see how I could get away with packing 3 months worth of liquids/gels in my checked baggage.  And that I'll be charged more to sit and drink something than to walk away with it.  Anyway, I've only just started so I'm eager for whatever tips I can pick up.  Right now I can't even quite ballpark a budget.  $3,000 (USD)?  $6,000?  $9,000?  I fear it will be prohibitively high, but there just has to be a way.

Thanks in advance for your help! 

hi again,
Santorini is one of the most expensive islands, I don't think is a good idea to go there, you'll find everything so expensive although is a stunning place, I do not recommended for such a long stay. It will cost you a small fortune and life is very loud indeed on this island!
The island of Milos(not far from santorini)belongs in the same group of islands/cyclades  you can find cheap accommodation such as small apartment even a small traditional house.
Google Milos islands greece to seach for accommodation, I might even try to help you from here to find a decent place.
I recommend you stay in a non expensive island but near the popular ones as you'll find that vacations and trips to those are scheduled on dalily basis.
You won't be able to travel with all the liquids you're describing in your hand bag but certainly you can pack them all in your suitcase.
Google MILOS GREEK ISLAND, THASOS GREEK ISLAND, IKARIA GREEK ISLAND,
any questions i am happy to help, I am in Athens at the moment.
best of luck
Kiki

Hi there,
We live on the Ioanian island of Corfu and wonder if you had considered this? We are both artists and travelled around a bit before finally choosing Corfu to settle,it's a popular holiday location and one of the more lush, green greek islands. We have a studio apartment with pool that we rent out in a typical greek village, google selfcateringcorfu.com to give you an idea of price,

I am also considering a 3 to 6 month stay in Greece, preferably on an island or at the beach.  I however will be taking 2 dogs, so I need to do a lot of research.  I would be most grateful if you could share info with me.  I do not plan to leave until the fall.