Hola Travis,
The Flex Viewer is design to display services, as you can see within the following documentation. My suggestion on this would be to either convert the dwg to shapefile, feature class, etc. and then publish it, where you will be able to display the data within the Flex Viewer. If you have any further questions or issues, please submit a case with Support Services. Gracias, Travis!
ArcGIS Viewer for Flex
http://resources.arcgis.com/en/communities/flex-viewer/
Welcome to ArcGIS Viewer for Flex
http://resources.arcgis.com/en/help/flex-viewer/concepts/index.html#//01m300000015000000
Juan