After a lot of research, I have found that there is a fix similar to the PNG transparancy fix available to IE6, called CSS3Pie. The website to the solution is found here: http://css3pie.com
You will simply have to add behavior: url(path/to/pie_files/PIE.htc);
to your css anywhere you wish to implement rounded corners.
#myAwesomeElement {
border: 1px solid #999;
-webkit-border-radius: 10px;
-moz-border-radius: 10px;
border-radius: 10px;
behavior: url(path/to/pie_files/PIE.htc);
}