You could look at surfaces with maximal Picard rank. A surface is said to have maximal Picard rank if $H^{1,1}(X) \cap H^1(X, \mathbb{R})$ is spanned by curve classes. So the Kahler cone is the same as the ample cone, and you can compute the intersection pairing on $H^{1,1}$ by just intersecting curves with each other.

Now, this raises the question of whether there is a surface of general type with maximal Picard rank. Some googling found me the following:

Section 3 of [Quintic surfaces with maximum and other Picard numbers][1] shows that the Fermat sextic $w^6+x^6+y^6+z^6=0$ has maximal Picard rank. Apparently, this was originally computed by Beauville.

[Roulleau shows][2] that there are infinitely many Fano surfaces with maximal Picard rank, and gives a basis for the Neron-Severi group. (Confusingly, the [Fano surface][3] is not a Fano variety, but rather is of general type.)

<strike>A warning: If you look at [this paper][4] or [this one][5], they compute a lot of surfaces with maximal Picard. However, their method is to make a surface with an $A_n$ singularity and prove that its resolution has maximal Picard. Since this resolution has a $-2$ curve, it is automatically not of general type.</strike> Sorry. As Donu says, I was confused. $K$ will not be ample, but it will be big, and the varieties are general type. 

I was coming back to edit in one more idea -- if $C$ is the [Klein quartic][6], then I think that $C \times C$ has maximal Picard rank. But Jason Starr's comment is simpler than any of my suggestions.


  [1]: http://arxiv.org/abs/0812.3519
  [2]: http://arxiv.org/abs/1001.4855
  [3]: http://en.wikipedia.org/wiki/Fano_surface
  [4]: http://www.springerlink.com/content/g55236l64258418w/
  [5]: http://arxiv.org/abs/0812.3519
  [6]: http://en.wikipedia.org/wiki/Klein_quartic