  Let $M$ be a smooth compact connected surface, $P$ be either the real line
$\mathbb{R}$ or the circle $S^1$ and $f:M\to P$ be a Morse map. Denote by
$\mathcal{S}(f)$ and $\mathcal{O}(f)$ the corresponding stabilizer and orbit of
$f$ with respect to the right action of the group $\mathcal{D}(M)$ of
diffeomorphisms of $M$. In a series of papers the author described homotopy
types of $\mathcal{S}(f)$ and computed higher homotopy groups of
$\mathcal{O}(f)$. The present paper describes the structure of the remained
fundamental group $\pi_1 \mathcal{O}(f)$ for the case when $M$ is orientable
and differs from $2$-sphere and $2$-torus.
  The result holds as well for a larger class of smooth maps $f:M\to P$ having
the following property: the germ of $f$ at each of its critical points is
smoothly equivalent to a homogeneous polynomial $\mathbb{R}^2\to\mathbb{R}$
without multiple factors.
