I approached this by using an UpdatePanel in my webpart. Then I added a Button (more on this later) and a TextBox to the UpdatePanel.
I also have a JavaScript class which handles all of the logic for submitting a query after the user has paused while typing their query. It contains the event handler for the onkeyup event which is attached to the TextBox:
t.Attributes.Add("onkeyup", "javascript:oSearchClass.KeyUpEventHandler(event);");
I used setTimeout and clearTimeout to handle when the class should call a SubmitQuery function.
When SubmitQuery() is called, it makes the TextBox read only (so the user can't type anything while it is querying) and then "clicks" the button using click(). Since you're using a normal Button, you can handle the Button.click event like normal to re-query the list and display results.
If you don't want your user to see the button, you can simply put it inside a span WebControl that is hidden.