  We present a unified approach which gives completely elementary proofs of
three weighted sum formulae for double zeta values. This approach also leads to
new evaluations of sums relating to the harmonic numbers, the alternating
double zeta values, and the Witten zeta function. We discuss a heuristic for
finding or dismissing the existence of similar simple sums. We also produce
some new sums from recursions involving the Riemann zeta and the Dirichlet beta
functions.
