You can stop the information appearing in the address bar by using forms and the "POST" method to submit requests.
That doesn't help with links or GET requests from forms.
A hacky approach(not recommended, but would probably work) is to use a frameset in your application, and have the navigation happen on an internal frame.
Because the top level frameset doesn't get replaced, the url address of the browser remains what it was when originally loaded, even if you are sending parameters via the request string. The downside to that is you have a frameset, which is not optimal for many other reasons. However this approach would "solve" your problem of not displaying values in the address bar.
Note that these approaches don't hide the parameters from anybody really wanting to see them, just stops displaying them blatantly in the address bar.