I think similar questions always translate in the $L^p$ boundedness of a Fourier multiplier. In this case you want a Fourier multiplier which "exchanges the operator $D_1D_2D_3$ with the operator $D_1^3+D_2^3+D_3^3$". Consider the Fourier multiplier by the function 
\begin{equation}
m(\xi_1,\xi_2,\xi_3) = \frac{\xi_1\xi_2\xi_3}{|\xi_1|^3+|\xi_2|^3+|\xi_3|^3}. 
\end{equation}
Then as Grafakos, Modern Fourier Analysis, Thrid Edition, Example 6.2.6 notices, when the multiplier has this typ of homogeneity, i.e. in this case
\begin{equation}
m(\lambda \xi_1, \lambda\xi_2,\lambda\xi_3)=m(\xi_1,\xi_2,\xi_3),
\end{equation}by differentiation we get that for any multi index $\alpha \in \mathbb{Z}_{\geq 0}^3$ it holds that 
\begin{equation}
\lambda^{|\alpha|}\partial^\alpha m(\lambda \xi_1, \lambda\xi_2,\lambda\xi_3)=\partial^\alpha m(\xi_1,\xi_2,\xi_3),
\end{equation}
Then by picking $\lambda = (|\xi_1|^2+|\xi_2|^2+|\xi_3|^2)^{-1/2}$, it follows that 
\begin{equation}
|\partial^\alpha m( \xi_1, \xi_2,\xi_3)| \leq (|\xi_1|^2+|\xi_2|^2+|\xi_3|^2)^{-|\alpha|/2} (\sup_{|\xi|=1}|\partial^\alpha m (\xi)) | \leq  c_\alpha |\xi_1|^{-\alpha_1} |\xi_2|^{-\alpha_2} |\xi_3|^{-\alpha_3}.
\end{equation} 
Therefore $m$ satisfies the hypothesis of Corollary 6.2.5 of the same reference, and hence it is a bounded operator on $L^p(\mathbb{R^3}), p>1$.
Then you have that 
\begin{equation}
\Vert{D_1D_2D_3} u\Vert_p \leq C_p \sum_{i=1}^3 \Vert |D_i|^3 u \Vert _p =c_p\sum_{i=1}^3 \Vert H_i D_i^3 u \Vert _p \leq c_p \sum_{i=1}^3 \Vert  D_i^3 u \Vert _p, 
\end{equation}
where $|D_i|$ is the Fourier multiplier with symbol $|\xi_i|$ and $H_i$ is the Hilbert transform in the $i$-th coordinate.