\documentclass[a4paper,12pt]{article}
\usepackage{array}
\usepackage{mdwlist}
\makeatletter
\def\noVSpace{\@minipagetrue}
\newenvironment{tabItemize}{%
  \@minipagetrue%
  \begin{itemize*}%
}{\vspace{-\normalbaselineskip}%
  \end{itemize*}}
\makeatother

\begin{document}

\begin{tabular}{|p{2.5cm}|p{2.5cm}|p{2.5cm}|}   \hline
  \textbf{Heading} & \textbf{Big} and & \textbf{Bold} \\ \hline
  \begin{itemize}
    \item Meek
    \item Meek
    \item \textit{shape}
    \item \textsc{Meek}
  \end{itemize} 
&  
  \begin{itemize}
    \item mild
    \item mild
    \item \textit{and}
    \item \textsc{mild}
  \end{itemize} 
&  
  \begin{itemize}
    \item entry
    \item entry
    \item \textit{italic}
    \item \textsc{entry}
  \end{itemize} \\\hline
\end{tabular}

\bigskip
\begin{tabular}{|p{2.5cm}|p{2.5cm}|p{2.5cm}|}   \hline
  \textbf{Heading} & \textbf{Big} and & \textbf{Bold} \\ \hline
  \noVSpace
  \begin{itemize}
    \item Meek
    \item Meek
    \item \textit{shape}
    \item \textsc{Meek}
  \end{itemize} 
&  
  \noVSpace
  \begin{itemize}
    \item mild
    \item mild
    \item \textit{and}
    \item \textsc{mild}
  \end{itemize} 
&  
  \noVSpace
  \begin{itemize}
    \item entry
    \item entry
    \item \textit{italic}
    \item \textsc{entry}
  \end{itemize} \\\hline
\end{tabular}

\bigskip
\begin{tabular}{|p{2.5cm}|p{2.5cm}|p{2.5cm}|}   \hline
  \textbf{Heading} & \textbf{Big} and & \textbf{Bold} \\ \hline
  \begin{tabItemize}
    \item Meek
    \item Meek
    \item \textit{shape}
    \item \textsc{Meek}
  \end{tabItemize}
&  
  \begin{tabItemize}
    \item mild
    \item mild
    \item \textit{and}
    \item \textsc{mild}
  \end{tabItemize}
&  
  \begin{tabItemize}
    \item entry
    \item entry
    \item \textit{italic}
    \item \textsc{entry}
  \end{tabItemize}\\\hline
\end{tabular}
\end{document}
