biblatex does this by default, and the biblatex-chicago package provides several different Chicago styles, see the manual. Note that the default backend for biblatex is biber, not bibtex, so the compilation sequence is
pdflatex file.tex
biber file.bcf
pdflatex file.tex
The bibkeys are also displayed before running biber and pdflatex the second time, whenever you'd get question marks with natbib/bibtex.
When switching from natbib to biblatex-chicago you need to make several changes in your document:
Replace \usepackage[round]{natbib} \bibliographystyle{chicago} in the preamble with
\usepackage[authordate]{biblatex-chicago}
\addbibresource{nameofyourbibfile.bib}
Replace \bibliography{nameofyourbibfile} in your document with \printbibliography.
Replace \citet with \textcite and \citep with \parencite. Use Find&replace in your editor to do this.
(Normally with plain biblatex you could have added natbib=true to the package options which would have made \citet and \citep available, but biblatex-chicago does not define this.)
Some questions with more information about biblatex:
Here is a short example:
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[authordate]{biblatex-chicago}
\addbibresource{biblatex-examples.bib}
\begin{document}
\textcite{aksin}
\textcite{notavailable}
\printbibliography
\end{document}
