Here is an example of how to do this with the use of some jQuery. Off the top of my head, so please forgive any typos.
<input id="field" name="field" type="text" value="" />
<g:link class="mylink" controller="somewhere" action="something">My link</g:link>
<script type="text/javascript">
jQuery(function(){
$("a.mylink").on("click", function(e) {
window.location.href = $(this).attr("href") + "?field=" + $("#field").val();
return false;
});
});
</script>