You can define a custom format that will show only the title.
In the Dashboard, go to Downloads, Configuration to define a Custom Output Format in the Add Format section.
The example code below is for custom output format #1 defined like this (the Name field could be any name):
Name: List with No Counts
Format:<a href="{url}" class="downloadlink" title="{version} downloaded {hits} times">{title}</a>
In your page you would list the files this way:
[downloads format="1"]
This should produce the same HTML output you have now except that the download count would not appear in the list.
I hope this helps.
Ray