Here is a sketch of an answer.  Suppose that $K$ is a simple closed curve in $M^3$.  I'll assume that $M$ is orientable, compact, and without boundary.  Let $V$ be a closed regular neighborhood of $K$; so $V \cong S^1 \times D^2$ is a solid torus.  Let $X$ be the closure of $M - V$; so $X$ is the _exterior_ of $K$.  Let $T = X \cap V$; so $T$ is a two-torus.  So $\partial X = \partial V = T$ and $M = X \cup_T V$.  Note that $T$ is a two-torus.  Let $D \subset V$ be a meridian disk; that is, a disk of the form $\lbrace \mbox{pt} \rbrace \times D^2$.

As Igor indicates, the map $\pi_1(T) \to \pi_1(X)$ induced by inclusion has a kernel if and only if there is a embedded disk in $E \subset X$ with boundary on $T$.  If $\partial D$ and $\partial E$ meet once then $K$ bounds a disk in $M$.