Wow. I searched for a while and didn't really find an obvious answer... so here it is:
Here's an excerpt from the original template. You need to remove the bits that I've put in bold and underlined.
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[files]
<div id=folderstats>%number-folders% folders, %number-files% files - Total: %total-size%</div>
<table cellpadding=5>
<th><a href="?sort=n">Filename</a>
<th><a href="?sort=s">Filesize</a>
<th><a href="?sort=t">Filetime</a>
<th><a href="?sort=d">Hits</a>
%list%
</table>
<div id=tools>
<a href="~files.lst?recursive">File list</a>
%archive%
</div>
[archive]
<br><a href="~folder.tar?recursive">Folder archive</a>
[protected]
<img src='/~img_lock'>
[file]
<tr><td>%new% %protected% <a href="%item-url%"><img src="/~img_file" /> %item-name%</a>%comment%<td align=right>%item-size%<td align=right>%item-modified%<td align=right>%item-dl-count%
[folder]
<tr><td>%new% %protected% <a href="%item-url%"><img src="/~img_folder" /> <b>%item-name%</b></a>%comment%<td align=center><i>folder</i><td align=right>%item-modified%<td align=right>%item-dl-count%
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Sorry that wasn't in "code" format--that doesn't show too well in some browsers and doesn't allow highlighting.
Explanation: The line removed from the [files] section just removes the header from the listing, but not the actual hits display. The pieces of the line removed in the [file] and the [folder] sections actually remove the hits counter display from the table.
Hope this helps.