You have to POST the URLRequest and assign the variables using URLVariables. What I really needed to do is something like this:
private function addGraphicToEasementLayer(graphic:Graphic):void
{
var easementJSON:Object = graphic.toJSON();
var easmentString:String = "["+JSONUtil.encode(easementJSON)+"]";
trace("the Easement is " + easmentString);
var request:URLRequest = new URLRequest('http://yourdomain/ArcGIS/rest/services/SharedMaps/GreenwayEasementsTest/FeatureServer/0/addFeatures'...
var urlVars:URLVariables = new URLVariables();
urlVars.f = "json";
urlVars.features = easmentString;
urlVars.gdbVersion = "";
urlVars.rollbackOnFailure = true;
request.data = urlVars;
request.method = URLRequestMethod.POST;
var loader:URLLoader = new URLLoader();
loader.load(request);
}
Works like a charm!
Hope this helps someone else.
Thanks, Tyler