A random evnironment on a countable metric group \(G\) is considered, and a random walk (RW) in the environment with bounded jumps is investigated. The transition probabilities from a point \(x\in G\) are described by a vector \(p(x)\in\mathbb{R}^{|W|}\) where \(W=W(x)\subset G\), and \(|W |= \text{const} <\infty\). The set \(\{p(x)\mid x\in G\}\) is supposed to consist of i.i.d. random vectors. Such a model for \(G=\mathbb{Z}\) was studied by \textit{E. S. Key} [Ann. Probab. 12, 529-560 (1984; Zbl 0545.60066)] and by \textit{A. V. Letchikov} [Sov. Math., Dokl. 39, No. 1, 15-18 (1989); translation from Dokl. Akad. Nauk SSSR 304, No. 1, 25-28 (1989; Zbl 0678.60060)], where a criterion for the RW to be transient was given. In the present paper a sufficient condition for this property is found for arbitrary \(G\), and the problem is reduced to the case \(G=\mathbb{Z}\). Particular examples of the group \(\mathbb{Z}^d\), free groups, and the free product of infinitely many cyclic groups of second order are considered.