Is there some way put boxes around matrix elements without too much effort?

Here is TikZ proposal using fit. I've wrapped the code in a command I've called mymatrixbox. It takes 4 inputs which define the boundaries of the box using the cell coordinates (it also takes 1 optional input which allows you some control over the default style).

The way it works is like this, say you want to draw a box from (1,1) to (3,1), then use \mymatrixbox{1}{1}{3}{1}. To draw a box around a single cell then just repeat the cell coordinates - e.g. \matrixbox{2}{3}{2}{3}. If I wanted to draw a box around the entire matrix the code for that would be \mymatrixbox{1}{1}{3}{3}.

\documentclass[margin=0.5cm]{standalone}

\usepackage{tikz}
\usetikzlibrary{matrix,fit}
\pgfkeys{tikz/mymatrix/.style={matrix of math nodes,inner sep=0pt,row sep=0em,column sep=0em,nodes={inner sep=6pt}}}

\newcommand*\mymatrixbox[5][]{\node [fit= (m-#2-#3) (m-#4-#5)] [draw=blue,thick,dashed,rounded corners,inner sep=-2pt,#1] {};}

\begin{document}

\begin{tikzpicture}[baseline=0cm]
    \matrix [mymatrix] (m)  
    {
    1 & 0 & 0 \\ 
    0 & 1 & 0 \\
    0 & 0 & 1 \\
    };

\mymatrixbox{1}{1}{3}{1}
\mymatrixbox{3}{2}{3}{3}
\mymatrixbox[red]{2}{3}{2}{3}

\end{tikzpicture}

\end{document}

A simple solution with pstricks and a plain matrix environment from amsmath:

\documentclass[11pt, svgnames]{article}
\usepackage{amssymb, amsmath, array}
\usepackage{pst-node} 
\usepackage{auto-pst-pdf} 

\begin{document}

 \begin{equation*}
        \begin{postscript}
        \psset{nodesep=5pt, boxsize=0.7em, linearc=.1, framearc=0.2, linestyle=dashed, dash= 4pt 2pt, linecolor=LightBlue}
        \setlength{\arraycolsep}{6pt}
        \setlength{\extrarowheight}{3pt}
            \begin{matrix}
                \Rnode{A}{1}& 0 & 0 \\
                0 &1 & \rnode{E}{0} \\
             \Rnode{B}{0}& \Rnode{C}{0} & \Rnode{D}{1}
            \end{matrix}
        \ncbox{A}{B}
        \ncbox{C}{D}
        \fnode[framesize = 0.56 0.50](E){F}
        \end{postscript}
        \end{equation*}

\end{document}