  In this paper, we provide new discrete uniformization theorems for bounded,
$m$-connected planar domains. To this end, we consider a planar, bounded,
$m$-connected domain $\Omega$ and let $\bord\Omega$ be its boundary. Let
$\mathcal{T}$ denote a triangulation of $\Omega\cup\bord\Omega$. We construct a
\emph{new} decomposition of $\Omega\cup\bord\Omega$ into a finite union of
quadrilaterals with disjoint interiors. The construction is based on utilizing
a {\it pair} of harmonic functions on ${\mathcal T}^{(0)}$ and properties of
their level curves. In the sequel \cite{Her3} it will be proved that a
particular discrete scheme based on these theorems converges to a conformal
map, thus providing an affirmative answer to a question raised by Stephenson
\cite[Section 11]{Steph}.
