It is easy to visualize some Morse functions on surfaces (such as the torus) via the height function, but this seemingly doesn't work for 3-manifolds. I am looking for an explicit one on the 3-torus $\mathbb{T}^3$. In paritcular:
It is true that every nice-enough 3-manifold admits a self-indexing Morse function $f:M\to[0,3]$ (with unique maximum and minimum), and from this we get a Heegaard diagram, with splitting surface $\Sigma=f^{-1}(\frac{3}{2})$. Now apparently, the converse also holds, so that given a Heegaard decomposition we can read off a self-indexing Morse function.
With the known Heegaard decomposition of $\mathbb{T}^3$ (handlebodies $N$ and $\mathbb{T}^3-N$ for a small neighborhood $N$ about the generating 1-complex $S^1\cup S^1\cup S^1\subset\mathbb{T}^3$, and splitting surface $\partial N$), what is the corresponding Morse function? I can't write one down.

It is easy to visualize some Morse functions on surfaces (such as the torus) via the height function, but this seemingly doesn't work for 3-manifolds. I am looking for an explicit one on the 3-torus $\mathbb{T}^3$. In paritcular:
It is true that every nice-enough 3-manifold admits a self-indexing Morse function $f:M\to[0,3]$ (with unique maximum and minimum), and from this we get a Heegaard diagram, with splitting surface $\Sigma=f^{-1}(\frac{3}{2})$. Now apparently, a converse also holds, so that given a Heegaard decomposition we can read off a self-indexing Morse function (edit: perhaps multiple Morse functions, as per Ryan Budney's comment).
With the known Heegaard decomposition of $\mathbb{T}^3$ (handlebodies $N$ and $\mathbb{T}^3-N$ for a small neighborhood $N$ about the generating 1-complex $S^1\cup S^1\cup S^1\subset\mathbb{T}^3$, and splitting surface $\partial N$), what is the corresponding Morse function? I can't write one down.