The study of the relationship between a ring $R$ and its subring of invariants $R^G$ under the action of a group $G$, \emph{invariant theory} for short, is a classical algebraic theme. Recently, a particular type of group action on a ring, the \emph{multiplicative action}, has attracted much attention. This action arises from \emph{$G$-lattices}, that is, from free abelian groups $A$ of finite rank $n$ on which the group $G$ acts by automorphisms. The $G$-action on $A$ extends uniquely to an action on the group algebra $R=k[A]$. In explicit terms, after choosing a $\Z$-basis, $A$ can be viewed as $\mathbb{Z}^n$, the $G$-action on $A$ is given by a homomorphism $G\to\mbox{GL}_n(\Z)$, and the group algebra $R=k[A]$ becomes the Laurent polynomial algebra $k[x_1,x_1^{-1},\ldots,x_n, x_n^{-1}]$. I will discuss the structure of the ring of multiplicative invariants, $k[A]^G$, when $G$ is a reflection group, and describe an algorithm for computing its algebra generators.