Hi Rick,
First read
http://www.viart.com/new_template-how_to.html to see the advised way of changing templates.
Copy "templates/user/block_products_list.html" to your new folder. This means you have the original if things go "pear-shaped".
The php engine uses <!-- begin xxxxxx --> and <!-- end xxxxxx --> markers to link the php to the html so these markers should not be removed otherwise the php will generate an error.
So examine the markup and remove as required.
( I wont say exactly which bits as it would take all the fun out of it
)