How do we do that?

Simple! We need to set a blank space for text in SubMenuText property and use the style folder for submenu icons:
<tm:TextMenu runat="server" id="tm1"
    StyleFolder="Styles/SubmenuIcon"
    SubMenuText=" ">
</tm:TextMenu>

Note: Property SubMenuText specifies the text that is shown by the Menu Item when it has submenu



This is how the menu looks now:



If you want to use a different icon, simply create your own and change the url in style.css file for classes with #tm_submenuCell to point to your file:
    .menuItem #tm_submenuCell
    {
        ....
        background-image: url('myImage.gif');
        ....
    }