The first thing that came to my mind was a javascript solution, but here is a php solution:
[PHP]
<?php

if (date('Y-m-d',strtotime('6/1/2005')) >= date('Y-m-d',time())) {
echo "<br />June 1: Game 1, blah, blah, blah";
}

if (date('Y-m-d',strtotime('6/15/2005')) >= date('Y-m-d',time())) {
echo "<br />June 15: Game 2, blah, blah, blah";
}

if (date('Y-m-d',strtotime('7/03/2005')) >= date('Y-m-d',time())) {
echo "<br />July 3: Game 3: blah, blah, blah";
}

?>
[/PHP]

You could also do a DHTML solution where you place each game within a DIV or SPAN. Give each DIV or SPAN an id equal to the game date. Then have a javascript fire onload that looks at all the DIVs or SPANs and if date is less than today, set (element).style.display="none";

PHP solution means you don't have to worry about any browser compatibility issues.