By an Euler walk in a 3-uniform hypergraph H we mean an alternating sequence v₀,e₁,v₁,e₂,v₂,…,v_m-1,e_m,v_m of vertices and edges in H such that each edge of H appears in this sequence exactly once and v_i-1,v_i∈e_i, v_i-1¬=v_i , for every i=1,2,…,m. This concept is a natural extension of the graph theoretic notion of an Euler walk to the case of 3-uniform hypergraphs. We say that a 3-uniform hypergraph H is strongly connected if it has no isolated vertices and for each two edges e and f in H there is a sequence of edges starting with e and ending with f such that each two consecutive edges in this sequence have two vertices in common. In this paper we give an algorithm that constructs an Euler walk in a strongly connected 3-uniform hypergraph (it is known that such a walk in such a hypergraph always exists). The algorithm runs in time O(m), where m is the number of edges in the input hypergraph.