<ul id="hdrnav">
                <li><a href="/apex/billing" id="billnav">Billing</a></li>
                <li><a href="/apex/deposits" id="depositsnav">Deposits</a></li>  
                <li><a href="/apex/checks" id="checksnav">Checks</a></li>   

            </ul>

Now I want to be able to render these

I know we can check a user's access to a record using something like:

{!$ObjectType.objectname.accessible}

So how can I achieve something like not displaying the

Possible Suggestion/Solution #1

This is a bit off the cuff, but I think you can wrap it in a panel:

<apex:outputPanel rendered="{!$ObjectType.objectname.accessible}" layout="none"><!-- content --> </apex:outputPanel>

Would that work in your case?

Possible Suggestion/Solution #2

One option would be to compose them server side, then display with <apex:dataList> or <apex:repeat>. Useful especially if the list will grow in future?

To do that build some wrapper apex class (or even something as simple as Map<String, String>) that would hold the URL and display name. Anything you can do in VF with {!$ObjectType.Account.accessible} you can check with Account.sObjectType.getDescribe().isAccessible()

Possible Suggestion/Solution #3

Off the cuff, I believe you can overload the use of the <apex:variable> tag along with it's rendered flag to show/hide individual elements.

<apex:variable var="foo" rendered="{!$ObjectType.objectname.accessible} >
    <li> bar ... </li>
</apex:variable>

Possible Suggestion/Solution #4

If the VF level or Controller level is not an option (the two first things I would try) - then you could also potentially control it via jQuery. Give the LI's you want to hide a specific class, and then tie it back to a function which checks if the user has the permission, and then shows the LI (instead of the other way around).