Each instance of geometry (esri.geometry.Geometry) has a method called getExtent() which returns the extent of the geometry (polygon in your example). esri.geometry.Extent has a method called getCenter() that returns a point.
So you could do something like:
var myPolygonCenterLatLon = myPolygon.getExtent().getCenter();
You could alternatively calculate the centroids to polygons beforehand in ArcMap. Then host the layer which has a one-to-one relationship with the polygons.
This example finds the centre point of the polygon to label the coordinates
This uses the Geometry Service http://webhelp.esri.com/arcgisserver/9.3/dotNet/index.htm#geometry_service.htm