With the enumitem package you can use label={} to specify it on a per instance basis, or use \setlist[itemize,<n>]{label=<symbol>} to set it based on the nesting level, where:
<symbol> is to be used at the given nesting level <n>:

Here I have used the \checkmark from the amsfonts package, but you can use any symbol you like.
References:
Code:
\documentclass{article}
\usepackage{enumitem}
\usepackage{amsfonts}
\setlist[itemize,1]{label=$\times$}
\setlist[itemize,2]{label=$\checkmark$}
\setlist[itemize,3]{label=$\diamond$}
\setlist[itemize,4]{label=$\bullet$}
\begin{document}
\begin{itemize}[label={$\bullet$}]
\item foo
\end{itemize}
\begin{itemize}[label={\checkmark}]
\item bar
\end{itemize}
\begin{itemize}
\item foo
\begin{itemize}
\item bar
\begin{itemize}
\item abc
\begin{itemize}
\item def
\end{itemize}
\end{itemize}
\end{itemize}
\end{itemize}
\begin{itemize}
\item bar
\end{itemize}
\end{document}
Without the enumitem package you could use \renewcommand{\labelitem<n>}{<symbol>} to redefine the marker, where <n> is a roman numeral (i, i, iii, or iv) representing the nesting depth of itemize. The following yields a similar result to above:
\documentclass{article}
\usepackage{amsfonts}
\begin{document}
\renewcommand{\labelitemi}{$\bullet$}
\begin{itemize}
\item foo
\end{itemize}
\renewcommand{\labelitemi}{$\checkmark$}
\begin{itemize}
\item bar
\end{itemize}
\renewcommand{\labelitemi}{$\times$}
\renewcommand{\labelitemii}{$\checkmark$}
\renewcommand{\labelitemiii}{$\diamond$}
\renewcommand{\labelitemiv}{$\bullet$}
\begin{itemize}
\item foo
\begin{itemize}
\item bar
\begin{itemize}
\item abc
\begin{itemize}
\item def
\end{itemize}
\end{itemize}
\end{itemize}
\end{itemize}
\begin{itemize}
\item bar
\end{itemize}
\end{document}