That depends on what you mean by 'explicit'. For example, in Some examples of special Lagrangian tori (Adv. Theor. Math. Phys., vol. 3 no. 1 (1999), pp. 83–90, also available at arXiv:math/9902076), I point out some very elementary examples of special Lagrangian tori in certain Calabi-Yau manifolds that occur as hypersurfaces in complex projective space. All of these are constructed as real slices of smooth hypersurfaces defined over the reals, which is a known method for constructing special Lagrangian tori.

By McLean's deformation theorem, when the hypersurface has complex dimension $3$, each such torus lies in a $3$-parameter family of special Lagrangian tori in the Calabi-Yau, though, as far as I know, no one knows how to write down the family explicitly. One certainly expects that, outside some locus of smaller dimension, this $3$-parameter family foliates the entire Calabi-Yau, though, as far as I know, this has never been proved rigorously.

That's an old paper by now, though, and someone may have constructed a more explicit example in the meantime that I don't know about.

Added Remark: Also, I should say that it depends on what you mean by 'CY3' as well. If by this, you mean a compact Ricci-flat Kähler $3$-fold, then, of course, there are products, such as the product of $3$ elliptic curves or the product of an elliptic curve and a K3 surface. In this cases, one can construct these more-or-less explicitly.

That depends on what you mean by 'explicit'. For example, in Some examples of special Lagrangian tori (Adv. Theor. Math. Phys., vol. 3 no. 1 (1999), pp. 83–90, also available at arXiv:math/9902076), I point out some very elementary examples of special Lagrangian tori in certain Calabi-Yau manifolds that occur as hypersurfaces in complex projective space. All of these are constructed as real slices of smooth hypersurfaces defined over the reals, which is a known method for constructing special Lagrangian tori.

By McLean's deformation theorem, when the hypersurface has complex dimension $3$, each such torus lies in a $3$-parameter family of special Lagrangian tori in the Calabi-Yau, though, as far as I know, no one knows how to write down the family explicitly. One certainly expects that, outside some locus of smaller dimension, this $3$-parameter family foliates the entire Calabi-Yau, though, as far as I know, this has never been proved rigorously. One piece of evidence in favor of this, is that, for the corresponding real slices of Calabi-Yau hypersurfaces of complex dimensions $1$ and $2$, the corresponding family of special Lagrangian tori do, indeed, foliate the hypersurface and generate a fibration of exactly the desired kind.

That's an old paper by now, though, and someone may have constructed a more explicit example in the meantime that I don't know about.

Added Remark: Also, I should say that it depends on what you mean by 'CY3' as well. If by this, you mean a compact Ricci-flat Kähler $3$-fold, then, of course, there are products, such as the product of $3$ elliptic curves or the product of an elliptic curve and a K3 surface. In these cases, one can construct the special Lagrangian fibrations more-or-less explicitly.

That depends on what you mean by 'explicit'. For example, in Some examples of special Lagrangian tori (Adv. Theor. Math. Phys., vol. 3 no. 1 (1999), pp. 83–90, also available at arXiv:math/9902076), I point out some very elementary examples of special Lagrangian tori in certain Calabi-Yau manifolds that occur as hypersurfaces in complex projective space. All of these are constructed as real slices of smooth hypersurfaces defined over the reals, which is a known method for constructing special Lagrangian tori.

By McLean's defformation theorem, each such torus lies in a $3$-parameter family of special Lagrangian tori in the Calabi-Yau, though, as far as I know, no one knows how to write down the family explicitly. One certainly expects that, outside some locus of smaller dimension, this $3$-parameter family foliates the entire Calabi-Yau, though, as far as I know, this has never been proved rigorously.

That's an old paper by now, though, and someone may have constructed a more explicit example in the meantime that I don't know about.

That depends on what you mean by 'explicit'. For example, in Some examples of special Lagrangian tori (Adv. Theor. Math. Phys., vol. 3 no. 1 (1999), pp. 83–90, also available at arXiv:math/9902076), I point out some very elementary examples of special Lagrangian tori in certain Calabi-Yau manifolds that occur as hypersurfaces in complex projective space. All of these are constructed as real slices of smooth hypersurfaces defined over the reals, which is a known method for constructing special Lagrangian tori.

By McLean's deformation theorem, when the hypersurface has complex dimension $3$, each such torus lies in a $3$-parameter family of special Lagrangian tori in the Calabi-Yau, though, as far as I know, no one knows how to write down the family explicitly. One certainly expects that, outside some locus of smaller dimension, this $3$-parameter family foliates the entire Calabi-Yau, though, as far as I know, this has never been proved rigorously.

That's an old paper by now, though, and someone may have constructed a more explicit example in the meantime that I don't know about.

Added Remark: Also, I should say that it depends on what you mean by 'CY3' as well. If by this, you mean a compact Ricci-flat Kähler $3$-fold, then, of course, there are products, such as the product of $3$ elliptic curves or the product of an elliptic curve and a K3 surface. In this cases, one can construct these more-or-less explicitly.