I am seeing the problem by looking at your page. Please try to switch the Link Display format from Table to Unordered List in the Link Library admin panel. I am seeing from the output that you seem to have li tags before and after each link item but have the output configured for table mode. Thus, the ul tags are missing. Other browsers work around that code error but not IE8.
Let me know if this works and close out the issue if it does.
I have a similar problem with IE: on http://www.undefeatedmag.com except it is with and image slider. In IE the do not slide they are just lined up on top of each other like separate post. Is there a plugin that will fix this?