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We obtain a noncommutative multivariable analogue of Louhichi and Olofssoncharacterization of Toeplitz operators with harmonic symbols on the weightedBergman space $A_m({\bf D})$, as well as Eschmeier and Langendorfer extensionto the unit ball of ${\bf C}^n$. All our results are proved in the more generalsetting of noncommutative poly-hyperballs ${\bf D_n^m}(H)$, ${\bf n,m}\in {\bfN}^k$, and are used to characterize the bounded free $k$-pluriharmonicfunctions with operator coefficients on poly-hyperballs and to solve theassociated Dirichlet extension problem. In particular, the results hold for thereproducing kernel Hilbert space with kernel  $$  \kappa_{\bf m}(z,w):=\prod_{i=1}^k \frac{1}{(1-\bar z_i w_i)^{m_i}},\qquadz,w\in {\bf D}^k,  $$  where $m_i\geq 1$. This includes the Hardy space, the Bergman space, and theweighted Bergman space over the polydisk.

Brown-Halmos characterization of multi-Toeplitz operators associated with noncommutative poly-hyperballs