Great question! :)
The Turkish and Greek people (I mean the normal people, because racists, bigots and die-hard nationalists are sadly a minority in almost every country, we must ignore them though) can surely be friends. We have so many things in common after all. This is not just my opinion, but a fact. I'm certain most of us have already examples from our personal life, or from our friends, relatives, etc...
(Here is a personal story -well, kind of personal: My godmother and her family are Greeks from Istanbul. They had to leave, but her mum always tells me about their life there, the city, their friends, and especially, their next-door neighbours who sheltered them for a week in their house during the 1955 riots. When they left, the next-door lady was crying, because, as my godmother's mum tells me, "we were like sisters". After all these years, they are still in touch, and they have visited each other several times, in Athens and in Istanbul. And my godmother will marry a turkish guy in June this year).
So, on a people-to-people basis, we can be friends. As we have been for a long time.
Having said that (and at the risk of sounding bitter or ruining the mood of the other posters), there are unsolved issues between our countries that need to be settled. The minorities, the Cyprus problem, the rights of each country in the Aegean etc have to be resolved once and for all, by mutual co-ooperation and on the basis of respect to international law and human rights. If we solve these problems, apart from preventing the loss of human lives (first and foremost) and achieving a closer alliance on a political level, it will be much easier for us to trust each other more and to let go of past grudges. And that, I think, is very important.
ADDED: I also agree with sismanusta.