A number of classical double sum identities have been derived by Euler, Nielsen, Ramanujan and others. A natural extension of these sums is to consider triple sums of the form \[ C_{p,q,r}= \sum_{j=2}^ \infty \sum_{k=1}^{j-1} \sum_{i=1}^{k-1} j^{-p} k^{-q} i^{-r} \qquad (p\geq 2,\;q,r\geq 1). \] Various relations between these sums are derived which are then used to obtain explicit representations of these series in terms of the Riemann zeta function and products of it.