Not sure how I didn't figure this out the first time around, but here's the solution:
<foreach param="filename" absparam="absfilename" target="echo-file">
<fileset dir="${project.basedir}">
<include name="*" />
<exclude name="/" />
<exclude name="*.*" />
</fileset>
</foreach>
Excluding "/" in the fileset removes the root directory.