The package offers a set of starred versions of \alph, \Alph,
\arabic, \roman and \Roman, without argument, for the current counter in enumerate, so you can use \arabic* to get the Arabic representation. A little example:
\documentclass{article}
\usepackage{enumitem}
\begin{document}
\begin{enumerate}[label={\bfseries Case \arabic*:}]
\item First.
\end{enumerate}
\end{document}
Or define a new dedicated list:
\documentclass{article}
\usepackage{enumitem}
\newlist{ecases}{enumerate}{1}
\setlist[ecases,1]{label={\bfseries Case \arabic*:}}
\begin{document}
\begin{ecases}
\item First.
\end{ecases}
\end{document}

As an additional useful feature, enumitem has a provision to handle user defined counters as well, using \AddEnumerateCounter or \AddEnumerateCounter* The following example illustrates a possible use of the non-starred variant (the example is a variation of one given in the package documentation):
\documentclass{article}
\usepackage{enumitem}
\makeatletter
\def\ctext#1{\expandafter\@ctext\csname c@#1\endcsname}
\def\@ctext#1{\ifcase#1\or First\or Second\or Third\or
Fourth\or Fifth\or Sixth\fi}
\AddEnumerateCounter{\ctext}{\@ctext}{Second}
\makeatother
\newlist{steps}{enumerate}{1}
\setlist[steps,1]{label={\ctext*:},align=left}
\begin{document}
\begin{steps}
\item do this.
\item do that.
\item don't do this.
\end{steps}
\end{document}

\usepackage[shortlabels]{enumitem}. Then the exact same syntax will be available (plus all the extra options fromenumitem). – Manuel Apr 23 '14 at 18:55