The physicists (see e.g. this paper of Aganagic and Vafa) will write the mirror as a threefold $X$ which is an affine conic bundle over the holomorphic symplectic surface $\mathbb{C}^{\times}\times \mathbb{C}^{\times}$ with discriminant a Seiberg-Witten curve $\Sigma \subset \mathbb{C}^{\times}\times \mathbb{C}^{\times}$. In terms of the affine coordinates $(u,v)$ on $\mathbb{C}^{\times}\times \mathbb{C}^{\times}$, the curve $\Sigma$ is given by the equation
$$
\Sigma : \ u + v + a uv^{-1} + 1 = 0,
$$
and so $X$ is the hypersurface in $\mathbb{C}^{\times}\times \mathbb{C}^{\times} \times \mathbb{C}^2$ given by the equation
$$
X : \ xy = u + v + a uv^{-1} + 1.
$$
From geometric point of view it may be more natural to think of the mirror not as an affine conic fibration over a surface but as an affine fibration by two dimensional quadrics over a curve. The idea will be to start with the Landau-Ginzburg mirror of $\mathbb{P}^{1}$, which is $\mathbb{C}^{\times}$ equipped with the superpotential $w = s + as^{-1}$ and to consider a bundle of affine two dimensional quadrics on $\mathbb{C}^{\times}$ which degenerates along a smooth fiber of the superpotential, e.g. the fiber $w^{-1}(0)$. In this setting the mirror will be a hypersurface in $\mathbb{C}^{\times}\times \mathbb{C}^{3}$ given by the equation
$$
xy - z^2 = s + as^{-1}.
$$
Up to change of variables this is equivalent to the previous picture but it also makes sense in non-toric situations. Presumably one can obtain this way the mirror of a Calabi-Yau which is the total space of a rank two (semistable) vector bundle of canonical determinant on a curve of higher genus.
Since no one else has tried to answer, I'll take a shot. It seems to me that there are threads of ideas in this story that in the very distant future might be woven together to give a possible answer.
To begin, we should note that there seems to be a general idea, discussed in this mathoverflow question, Deformation quantization and quantum cohomology (or Fukaya category) -- are they related?. That one could define the Fukaya category as modules over a deformation quantization of $C^{\infty}(X)$ corresponding to the symplectic form $\omega$.
The basic idea is that in two naive respects this category of modules behaves a lot like the Fukaya category. Firstly, the Hochschild cohomology of the deformation quantization is almost by definition the Poisson cohomology of the symplectic form $\omega$, which in turn is known to be isomorphic to $H^*(X)((t))$. As an equation:
$$HH^*(A_\omega,A_\omega) \cong H^*(X)((t)) $$
Second, one can define a reasonable notion of modules with support on a Lagrangian submanifold and for any Lagrangian L, produce canonical holonomic modules supported there. One can compute that $$Ext(M_L,M_L) \cong H^*(L)((t))$$ There is some hope that one can put in the instanton corrections in a formal algebraic way and a fair amount of work has been done in this direction.
This story works best so far for the Fukaya category of $T^*X$ where the deformation quantization is roughly the algebra of differential operators. This is related to more work than I could competently summarize. I'll just mention, work of Nadler and Zaslow, Tsygan and Tamarkin. This approach is used by Kapustin and Witten to incorporate co-isotropic branes into the Fukaya category in their famous study of the Geometric Langlands. There, they are after some enlargement of Nadler's infinitesimal Fukaya category of $T^*(X)$. Note however that this not the same Fukaya category(the wrapped Fukaya category) that one studies in the context of mirror symmetry, but perhaps things will work better in the compact case if that is ever put on firm ground.
This was all a prelude to say that deformation quantization places you firmly in the land of non-commutative geometry anyways. Things like differential operators for non-commutative rings can make sense http://www.springerlink.com/content/r0rqguawu1960qxy/. I've never really looked at Van Den Bergh's work, but perhaps the passage from the sheaf of algebraic functions to the sheaf $C^\infty(X)$ is another stumbling point. One of Maxim's Kontsevich's ideas (see his Lefschetz lecture notes http://www.ihes.fr/~maxim/TEXTS/Kontsevich-Lefschetz-Notes.pdf) is that for any saturated dg-algebra there should maybe exist some nuclear algebra which bears the same formal relationship as the algebra of algebraic functions and smooth functions.
Best Answer
An answer to this question depends on what you mean by "mirror". As to topological mirror symmetry, the Batyrev-Borisov toric construction is the easiest and the most standard. Other examples would be Borcea-Voisin CY3s. If you want to do more, for example instanton computation etc, you need to work on CY3s with small $h^{1,1}$ (or mirrorly $h^{2,1}$) so that you can compute the mirror map, period integral etc.
CY3s with $h^{1,1}=1$ are typically complete intersections of Grassmannians. Batyrev, Ciocan-Fontanine, Kim and van Straten construct mirror manifolds (with mild singularity) of such CY3s via toric degeneration in this paper. The basic idea is easy to understand; they degenerate the ambient Grassmannians to toric varieties and apply the Batyrev-Borisov toric construction. So the essential part is toric in this case, too.
Apart from toric examples, the best known example would be Rodland's pfaffian-7 CY3. In this paper, he constructs a smooth mirror manifold of the pfaffian-7 CY3 by an orbifold method, i.e. taking a special 1-parameter family and take quotient by a finite group. The interesting observation is that the mirror manifold is also mirror of the complete intersection CY3 of Gr(2,7). Later Borisov and Caldararu prove that Rodland's CY3 and the complete intersection CY3 of Gr(2,7) are derived equivalent as HMS indicates.
Recently Kanazawa constructs several new pfaffian CY3 with $h^{1,1}=1$ in this paper, partially solving van Enckevort and van Straten's conjecture. Mirror manifolds (with mild singularity) are also constructed by an orbifold method. Kanazawa's pfaffian CY3s are interesting in the sense that the mirror manifolds have two large complex structure limits (similar to Rodland's case). This kind of phenomenon is also found by Hosono and Takagi in this paper. Their mirror symmetry is based on the standard Batyrev-Borisov toric method (with mild modification), but a non-trivial Fourier-Mukai partner comes into play.
Apart from toric examples, there seems no standard way to construct mirror manifolds for a given smooth compact CY3. Rodland's and Kanazawa's construction is some what an art. People have been working hard to find an intrinsic and systematic mirror construction. SYZ conjecture is one of such attempts.