The tufte classes were designed for this purpose (amongst others), so you could give them a try. However, as I see from the comments to the question, you want to stick to amsart. As a personal note, I don't understand this: AMS classes were designed for specific typographic needs which don't include what you want, and tufte classes already offer you what you want.
Anyway, if you want to (ab)use the AMS classes, you'll have to do some work. One possibility is to take advantage of the space reserved for marginal notes and use the geometry package to change the page layout adding some additional space to the width reserved for these marginal notes. With the marginnote package, you can use \marginnote to typeset the images in the marginal notes area, using \captionof (from the caption package) to provide captions. A little example that could give you a starting point:
\documentclass[oneside]{amsart}
\usepackage[rmargin=3cm,textwidth=11cm,marginparwidth=6cm]{geometry}
\usepackage{graphicx}
\usepackage{marginnote}
\usepackage{caption}
\usepackage{lipsum}
\captionsetup{justification=raggedright}
\newcommand\MarginFigure[4][width=4cm]{%
\marginnote{%
\begin{minipage}{\linewidth}
\centering
\includegraphics[#1]{#2}
\captionof{figure}{#3}
\label{#4}
\end{minipage}}}
\reversemarginpar
\begin{document}
\lipsum[1]
\MarginFigure{example-image-a}{a test figure}{fig:testa}
\lipsum[3-4]
\MarginFigure{example-image-b}{a test figure}{fig:testb}
\lipsum[1-3]
\MarginFigure{example-image-c}{a test figure}{fig:testb}
\lipsum[2-4]
\end{document}

tuftedocument classes – cmhughes Aug 04 '13 at 16:35amsartdocument that I already have set up, instead of using a new\documentclassthat requires completely different commands. Thanks for your input, though. – Randy Randerson Aug 04 '13 at 16:53