  In this paper, we study free k-pluriharmonic functions on noncommutative
regular polyballs. These regular polyballs have universal operator models
consisting of left creation operators acting on tensor products of full Fock
spaces. We introduce and determine the class of kmulti- Toeplitz operators
acting on these tensor products and show that the bounded free k-pluriharmonic
functions on regular polyballs are precisely the noncommutative Berezin
transforms of k-multi-Toeplitz operators. The Dirichlet extension problem on
regular polyballs is also solved. It is proved that a free k-pluriharmonic
function has continuous extension to the closed polyball if and only if it is
the noncommutative Berezin transform of a k-multi-Toeplitz operator in a
certain class, which we determine. We provide a Naimark type dilation theorem
for direct products of unital free semigroups, and use it to obtain a structure
theorem which characterizes the positive free k-pluriharmonic functions on the
regular polyball, with operator-valued coefficients. We define the
noncommutative the Berezin (resp. Poisson) transform of a completely bounded
linear map on the C*-algebra generated by the universal operator model and give
necessary an sufficient conditions for a function to be the Poisson transform
of a completely bounded (resp. completely positive) map. In the last section of
the paper, we obtain Herglotz-Riesz representation theorems for free
holomorphic functions on regular polyballs with positive real parts, extending
the classical result as well as Korannyi-Pukanszky version in scalar polydisks.
