The distance or D-eigenvalues of a graph G are the eigenvalues of its distance matrix. The distance or D-energy ED(G) of the graph G is the sum of the absolute values of its D-eigenvalues. Two graphs G1 and G2 are said to be D-equienergetic if ED(G1) = ED(G2). Let F1 be the 5-vertex path, F2 the graph obtained by identifying one vertex of a triangle with one end vertex of the 3-vertex path, F3 the graph obtained by identifying a vertex of a triangle with a vertex of another triangle and F4 be the graph obtained by identifying one end vertex of a 4-vertex star with a middle vertex of a 3-vertex path. In this paper we show that if G is r-regular, with diam(G)? 2, and Fi,i = 1,2,3,4, are not induced subgraphs of G, then the k-th iterated line graph Lk(G) has exactly one positive D-eigenvalue. Further, if G is r-regular, of order n, diam(G)?2, and G does not have Fi,i=1,2,3,4, as an induced subgraph, then for k ?1, ED(Lk(G)) depends solely on n and r. This result leads to the construction of non D-cospectral, D-equienergetic graphs having same number of vertices and same number of edges.