  In this paper, we study a class of non-weight modules over two kinds of
algebras related to the Virasoro algebra, i.e., the loop-Virasoro algebras
$\mathfrak{L}$ and a class of Block type Lie algebras $\mathfrak{B(q)}$, where
$q$ is a nonzero complex number. We determine those modules whose restriction
to the Cartan subalgebra (modulo center) are free of rank one. We also provide
a sufficient and necessary condition for such modules to be simple, and
determine their isomorphism classes. Moreover, we obtain the simplicity of
modules over loop-Virasoro algebras by taking tensor products of some
irreducible modules mentioned above with irreducible highest weight modules or
Whittaker modules.
