Edit the template with the template editor. remove just the body parts of the templates.
Control Panel » Templates
Select "email_html.tpl" and click "Edit Selected Template"
FIND this
<tr>
<td {$styletdbody}>
{$posts[ind].body}
<br />
</td>
</tr>
Remove it.
The procedure is basically the same for the other email templates, but that is the one that 99.9% of your users will be getting.
Dont forget to create a backup before making any changes.