I'm using the ArcGIS Explorer Desktop (free viewer) - build 1750 (latest). It comes with several built in base maps, including OpenStreetMap. Is it possible to add to it OpenCycleMap: http://www.opencyclemap.org/ as well ?
An alternative is that ArcGIS Explorer Desktop supports WMS, so if you could find someone providing a service URL for an OpenCycleMap WMS then this could be added via Add Content | GIS Services then setting Server Type: WMS. I've just looked quickly for an OpenCycleMap WMS without success. If you are in Great Britain then the OSM-GB project could be worth contacting.