You can do something similar to this: http://help.arcgis.com/en/webapi/silverlight/samples/start.htm#SwitchMap but instead of RadioButton.Tag contain the URL string, have it contain resource key.XAML-code:
<Grid.Resources>
<esri:ArcGISTiledMapServiceLayer x:Key="MyTiledLayer" Url="http://services.arcgisonline.com/ArcGIS/rest/services/World_Street_Map/MapServer" />
<bing:TileLayer x:Key="MyBingLayer" LayerStyle="Road" Visible="True" ServerType="Production" Token="{StaticResource BingKeyString}" />
<esri:OpenStreetMapLayer x:Key="MyOpenStreetMapLayer" Style="Mapnik" />
</Grid.Resources>
Code-behind:
private void RadioButton_Click(object sender, RoutedEventArgs e)
{
MyMap.Layers.Clear();
Layer layer = this.LayoutRoot.Resources[(sender as RadioButton).Tag] as Layer;
MyMap.Layers.Add(layer);
}