Are there examples of Kähler manifolds whose Kähler cone can be described explicitly, say spanned by certain cohomology classes? As far as I know, Hirzebruch Surface has a complete description for its Kähler cone.

Flag manifolds G/B are nice: the K\"ahler cone is the positive Weyl chamber, with edges coming from the Poincar\'e duals of the Schubert divisors. 


Generalising the case of Hirzebrouch surface, you can say that toric varieties admit explicit description of Kahler cone. Also for each Fano variety its Kahler cone is polyhedral, i.e., it is spanned by a finite number of rays (but this does not mean, that the description is easy). If you leave the class of Fano varieties unpleasant things may start to happen. For example for a generic blow up of $\mathbb CP^2$ in $n\ge 10$ points the structure of Kahler cone it is still unknown (for $n<9$ we get Fano), this is related to Nagata conjecutre http://en.wikipedia.org/wiki/Nagata's_conjecture_on_curves Morrison's conjecture states that for a CalabiYau manifold the quotient of the Kahler cone by the group of isometries of the manifold is polyhedral. The conjecture was proved only for surfaces, there is a recent very nice paper of Burt Totaro on this topic "The cone conjecture for CalabiYau pairs in dimension two", http://arxiv.org/abs/0901.3361 


To Hirzebruch surfaces, you can add $\mathrm{CP}^2$, its $k$folds blowups, $1\leq k\leq 8$, and some irrational ruled surfaces. Related to this question is the determination of the symplectic cone. This is now understood for rational $4$manifolds, ruled $4$manifolds and their blowups, and also for some elliptic fibrations. There is a nice survey by TianJun Li of the relations between symplectic and Kahler cones for $4$manifolds (and complex surfaces). See arXiv:0805.2931. 


The cone of curves of K3 surfaces is described in this paper. 


Explicit description of a Kahler cone for all hyperkahler manifolds is here: http://arxiv.org/abs/1401.0479 (Rational curves on hyperkahler manifolds, Ekaterina Amerik, Misha Verbitsky) 

