Update: It is easy with KOMA-Script version 3.15 or newer. Just use the new command \RedeclareSectionCommand:
\RedeclareSectionCommand[counterwithin=part,tocnumwidth=2em]{chapter}
Note that the package chngcntr is not needed.

Code:
\documentclass[open=any]{scrbook}
\RedeclareSectionCommand[
counterwithin=part,
tocnumwidth=2.5em
]{chapter}
\begin{document}
\tableofcontents
\part{This is part 1}
\chapter{This is I.1}
\part{This is part 2}
\chapter{This is II.1}
\end{document}
You can use the package tocstyle which is part of the KOMA-Script-bundle.
\documentclass[openany]{scrbook}
\usepackage{tocstyle}
\usetocstyle{KOMAlike}
\usepackage{chngcntr}
\usepackage{blindtext}
\counterwithin{chapter}{part}
\begin{document}
\tableofcontents
\part{This is part 1}
\blinddocument\blinddocument
\part{This is part 2}
\blinddocument\blinddocument
\end{document}
You have to run the code several times to get:

Or you can use the option toc=flat:
\documentclass[openany,toc=flat]{scrbook}
\usepackage{chngcntr}
\usepackage{blindtext}
\counterwithin{chapter}{part}
\begin{document}
\tableofcontents
\part{This is part 1}
\blinddocument\blinddocument
\part{This is part 2}
\blinddocument\blinddocument
\end{document}
Run twice to get

\counterwithin*{chapter}{part}so that chapter numbers are not prepended by part number. – May 30 '14 at 11:15