  For a smooth algebraic variety $X$, a monodromic $D$-module on $X\times
\mathbb{C}$ is decomposed into a direct sum of some $D$-modules on $X$. We show
that the Hodge filtration of a mixed Hodge module on $X\times \mathbb{C}$ whose
underlying $D$-module is monodromic is also decomposed. Moreover, we show that
there is an equivalence of categories between the category of monodromic mixed
Hodge modules on $X\times \mathbb{C}$ and the category of ``gluing data''. As
an application, we endow the Fourier-Laplace transformation of the underlying
$D$-module of a monodromic mixed Hodge module with a mixed Hodge module
structure.
