  We initiate the study of weighted multi-Toeplitz operators associated with
noncommutative regular domains in B(H)^n. These operators are acting on the
full Fock space with n generators and have as symbols free pluriharmonic
functions. Several classical results from complex analysis concerning harmonic
functions have analogues in our noncommutative setting. In particular, we show
that the bounded free pluriharmonic functions are precisely those which are
noncommutative Berezin transforms of weighted multi-Toeplitz operators, and
solve the Dirichlet extension problem in this setting. Using noncommutative
Cauchy transforms, we provide a free analytic functional calculus for n-tuples
of operators, which extends to free pluriharmonic functions.
