There are some very good reasons why the majority of calculations are done for algebraic manifolds. Maybe the most naive reason is as follows: it is harder to solve PDEs than to draw lines through two points in a space. Somehow everyone knows that for two points in $\mathbb CP^n$ there is exactly one line that passes through them, and so a certain GW invariant equals $1$. From the point of view of symplectic geometry this is a completely non-trivial result. Indeed, if you take the standard symplectic structure $w$ on $\mathbb CP^n$ (coming from the Fubini-Study metric), take $J$ that is tamed by $w$ and try to make the calculation, you will first need to know that almost complex curves exist for $J$ locally (which is already non-trivial), then using different compactness arguments will need to homothopy $J$ to the standard complex structure and prove that GW invariant does not change during homothopy, and finally once you get the standard complex structure on $\mathbb CP^n$ perform the above elementary calculation to get $1$ line through any two points.
Gromov non-squeezing http://en.wikipedia.org/wiki/Nonsqueezing_theorem is also based on this idea, so all the above is a completely non-trivial task...
A different reason that majority of calculations are done in algebraic case is that up to recently majority of compact symplectic manifolds we constructed from algebraic pieces, by taking Gompf's sum. In this case again on can try first to calculate GW invariants for these pieces and then glue them. Also in small dimensions symplectic manifolds with sufficiently non-trivial GW invariants tend to be algebraic. For example, in dimension 4 it is conjectured that if a GW invariant of rational curves passing trough one point is non-zero then the 4-manifold is a unirulled or rational surface (in particular, it is algebraic).
If you want an explicit calculation for manifolds that are not Kahler, check the following article of McDuff "Hamiltonian S^1 manifolds are uniruled" http://arxiv.org/abs/0706.0675
. She proves non-vanishing of certain GW invariants for manifolds admitting Hamiltonian $S^1$ action and it is not hard to construct such non-Kahler manifolds.
More generally, Tien-Jun Li and Ruan argue in "Symplectic Birational Geometry" http://arxiv.org/abs/0906.3265 that rational GW invariants are responsible for birational geometry of symplectic manifolds.