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 $$K=C(S)=S\times K_1$$ K=C(S)=S×K1 , where S is a torus in a compact simple Lie group G and $$K_1$$ K1 is the semisimple part of K. Then, the associatedM-space is the homogeneous space $$G/K_1$$ G/K1 . 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 $$G/K_1$$ G/K1 is a g.o. metric. The analysis is based on properties of the isotropy representation $$\mathfrak {m}=\mathfrak {m}_1\oplus \cdots \oplus \mathfrak {m}_s$$ m=m1⊕⋯⊕ms of the flag manifold G / K [as $${{\mathrm{Ad}}}(K)$$ Ad(K) -modules].