(1) Choose a planar presentation of your link, approximately in the plane of the blackboard. Let $N$ be a tubular neighborhood of the link in this position.
(2) For each crossing $c$ of the presentation, add a 1-handle $T_c$ to $N$ which is perpendicular to the blackboard and connects the upper and lower parts of the crossing. Let $H$ be the union of $N$ and all the 1-handles $T_c$.
(3) $S^3 \setminus H$ is a handlebody. The corresponding set of Heegaard curves on $\partial H$ bijects with the complementary regions of the (flattened) planar diagram of the link.
(4) $H$ is, of course, also a handlebody. For the corresponding Heegaard curves take the surgery curves on $N$ (this involves a choice to make them disjoint from the attaching disks of the $T_c$) union the obvious small, disk-bounding curves on the boundary of each $T_c$. (Correction added later: Actually, if there are $k$ components of the link then one should omit $k-1$ of the $T_c$ curves. The omitted crossings should be minimal with respect to connecting the components of the link. Also, I'm assuming that the planar projection is a connected graph.)
