  We investigate mirror symmetry for toric Calabi-Yau manifolds from the
perspective of the SYZ conjecture. Starting with a non-toric special Lagrangian
torus fibration on a toric Calabi-Yau manifold $X$, we construct a complex
manifold $\check{X}$ using T-duality modified by quantum corrections. These
corrections are encoded by Fourier transforms of generating functions of
certain open Gromov-Witten invariants. We conjecture that this complex manifold
$\check{X}$, which belongs to the Hori-Iqbal-Vafa mirror family, is inherently
written in canonical flat coordinates. In particular, we obtain an enumerative
meaning for the (inverse) mirror maps, and this gives a geometric reason for
why their Taylor series expansions in terms of the K\"ahler parameters of $X$
have integral coefficients. Applying the results in \cite{Chan10} and
\cite{LLW10}, we compute the open Gromov-Witten invariants in terms of local
BPS invariants and give evidences of our conjecture for several 3-dimensional
examples including $K_{\proj^2}$ and $K_{\proj^1\times\proj^1}$.
