$(".makeItRich")
will not work because the textarea is inside your element's ShadowRoot, where JQuery will not find it. Also, the CSS is scoped to the element, so you must put your CSS links inside the template.
This worked when I tried it:
<link rel="import" href="../components/polymer/polymer.html">
<link rel="import" href="../components/jquery2-import/jquery2-import.html">
<script type="text/javascript" src="markitup/jquery.markitup.js"></script>
<script type="text/javascript" src="markitup/sets/default/set.js"></script>
<polymer-element name="markitup-element" attributes="rows cols value">
<template>
<style>
:host {
display: block;
}
</style>
<link rel="stylesheet" type="text/css" href="markitup/skins/markitup/style.css">
<link rel="stylesheet" type="text/css" href="markitup/sets/default/style.css">
<textarea id="rich" rows="{{rows}" cols={{cols}}" value="{{value}}"></textarea>
</template>
<script>
Polymer({
value: "",
rows: 25,
cols: 80,
domReady: function() {
$(this.$.rich).markItUp(mySettings);
}
});
</script>
</polymer-element>