  Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous
Riemannian spaces $(M=G/H,g)$ whose geodesics are orbits of one-parameter
subgroups of $G$. The corresponding metric $g$ is called a geodesic orbit
metric. We study the geodesic orbit spaces of the form $(G/H,g)$, such that $G$
is one of the compact classical Lie groups $\SO(n)$, $U(n)$, and $H$ is a
diagonally embedded product $H_1\times \cdots \times H_s$, where $H_j$ is of
the same type as $G$. This class includes spheres, Stiefel manifolds, Grassmann
manifolds and real flag manifolds. The present work is a contribution to the
study of g.o. spaces $(G/H,g)$ with $H$ semisimple.
