  P. Flajolet and B. Salvy \cite{FS1998} prove the famous theorem that a
nonlinear Euler sum $S_{i_1i_2\cdots i_r,q}$ reduces to a combination of sums
of lower orders whenever the weight $i_1+i_2+\cdots+i_r+q$ and the order $r$
are of the same parity. In this article, we develop an approach to evaluate the
cubic sums $S_{1^2m,p}$ and $S_{1l_1l_2,l_3}$. By using the approach, we
establish some relations involving cubic, quadratic and linear Euler sums.
Specially, we prove the cubic sums $S_{1^2m,m}$ and $S_{1(2l+1)^2,2l+1}$ are
reducible to zeta values, quadratic and linear sums. Moreover, we prove that
the two combined sums involving multiple zeta values of depth four
\[\sum\limits_{\left\{ {i,j} \right\} \in \left\{ {1,2} \right\},i \ne j}
{\zeta \left( {{m_i},{m_j},1,1} \right)}\quad {\rm and}\quad
\sum\limits_{\left\{ {i,j,k} \right\} \in \left\{ {1,2,3} \right\},i \ne j \ne
k} {\zeta \left( {{m_i},{m_j},{m_k},1} \right)} \] can be expressed in terms of
multiple zeta values of depth $\leq 3$, here $2\leq m_1,m_2,m_3\in \N$.
Finally, we evaluate the alternating cubic Euler sums ${S_{{{\bar 1}^3},2r +
1}}$ and show that it are reducible to alternating quadratic and linear Euler
sums. The approach is based on Tornheim type series computations.
