I finally found a solution.
I guess this solution is best for third party css files and perhaps another solution is needed for wanting to use multiple self-created css files.
define([
"dojo/_base/declare",
...
"dojo/text!WidgetName/widget/template/WidgetName.html"
], function (declare, ..., widgetTemplate) {
"use strict";
var $ = _jQuery.noConflict(true);
dom.addCss(require.toUrl("WidgetName/lib/path/to/file.css"));
...
...
You probably use a css include in the widget. The 6.x widgets don't use that include but an implicit include of a css file called "[WidgetName].css". Replace [WidgetName] with referenceSetSelector or xstyle or you widget name.