  We investigate homogeneous geodesics in a class of homogeneous spaces called
$M$-spaces, which are defined as follows. Let $G/K$ be a generalized flag
manifold with $K=C(S)=S\times K_1$, where $S$ is a torus in a compact simple
Lie group $G$ and $K_1$ is the semisimple part of $K$. Then the {\it associated
$M$-space} is the homogeneous space $G/K_1$. These spaces were introduced and
studied by H.C. Wang in 1954. We prove that for various classes of $M$-spaces
the only g.o. metric is the standard metric. For other classes of $M$-spaces we
give either necessary, or necessary and sufficient conditions, so that a
$G$-invariant metric on $G/K_1$ is a g.o. metric. The analysis is based on
properties of the isotropy representation $\mathfrak{m}=\mathfrak{m}_1\oplus
\cdots\oplus \mathfrak{m}_s$ of the flag manifold $G/K$ (as Ad$(K)$-modules)
and corresponding decomposition
$\mathfrak{n}=\mathfrak{s}\oplus\mathfrak{m}_1\oplus \cdots\oplus
\mathfrak{m}_s$ of the tangent space of the $M$-space $G/K_1$ (as
Ad$(K_1)$-modules).
