I'm assuming you meant homologically non-trivial. There are two very famous theorems of Gabai, which show up in quite a lot of places in 3-dimensional topology for constructing taut foliations (in particular the proofs of properties P and R).

Theorem(Gabai): Let $Y$ be an irreducible 3-manifold $\alpha \in H_2(Y)$, and let $\Sigma$ be a minimal genus surface representing $\alpha$ with no toroidal or spherical components. Then there exists a taut foliation on $Y$ so that $\Sigma$ is a disjoint union of leaves.

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Theorem(Gabai): Let $S^3_0(K)$ denote 0-surgery on a knot $K$ in $S^3$. Let $\Sigma$ be the closed surface formed by gluing a Seifert surface to the surgered in disk. Then there exists a taut foliation of $S^3_0(K)$ with $\Sigma$ as a leaf.

The original reference to these theorems are the three papers Foliations and the Topology of 3-Manifolds (I), II and III.

I'm assuming you meant homologically non-trivial. There are two very famous theorems of Gabai, which show up in quite a lot of places in 3-dimensional topology for constructing taut foliations (in particular the proofs of properties P and R).

Theorem(Gabai): Let $Y$ be an irreducible 3-manifold, a nonzero $\alpha \in H_2(Y)$, and let $\Sigma$ be a minimal complexity surface representing $\alpha$ with no toroidal or spherical components. Then there exists a taut foliation on $Y$ so that $\Sigma$ is a disjoint union of leaves.

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Theorem(Gabai): Let $S^3_0(K)$ denote 0-surgery on a knot $K$ in $S^3$. Let $\Sigma$ be the closed surface formed by gluing a minimal genus Seifert surface to the surgered in disk. Then there exists a taut foliation of $S^3_0(K)$ with $\Sigma$ as a leaf.

The original reference to these theorems are the three papers Foliations and the Topology of 3-Manifolds (I), II and III.