Tornheim-like series, harmonic numbers and zeta values

Explicit evaluations of the Tornheim-like double series in the form $$\sum _{n,m=1}^{\mathrm{\infty }}\frac{H_{n+m+s}}{nm(n+m+s)},s\in \mathbb{N}\cup \left\{0\right\}$$ and their extensions are given. Furthermore, series of the type $$\sum _{m=1}^{\mathrm{\infty }}\frac{2H_{2m+1}-H_m}{2m(2m+1)}$$ and some other Tornheim-like multiple series are evaluated in terms of the zeta values.