Thanks to Jessy for the info about it not being possible with the built-in Gizmos. Here's my hacky work-around:

1. I made a prefab to be the gizmo. I just used a cube scaled down nearly flat and positioned it at y=0 inside another game object. I added an alpha grid texture for bonus points.
2. Tag the prefab "EditorOnly". Given the other steps this is probably not strictly necessary, but won't hurt.

3. In the component that needed positioning, I did this:

   private static GameObject goGizmoPlane = null;
   private static string pathToGizmoPlane = "Assets/Gizmos/MyGizmoPlane.prefab";
   
   
   void OnDrawGizmosSelected()
   {
       if( Application.isEditor )
       {
           if( goGizmoPlane == null )
           {
               // create one
               GameObject prefab = Resources.LoadAssetAtPath( pathToGizmoPlane, typeof( GameObject ) ) as GameObject;
               if( !prefab )
               {
                   Debug.LogError( "Could not load " + pathToGizmoPlane );
               }
               else
               {
                   goGizmoPlane = GameObject.Instantiate( prefab ) as GameObject;
                   goGizmoPlane.hideFlags = HideFlags.HideAndDontSave;
               }
           }
           // position it
           if( goGizmoPlane.transform.position != transform.position )
               goGizmoPlane.transform.position = transform.position;
           if( goGizmoPlane.transform.rotation != transform.rotation )
               goGizmoPlane.transform.rotation = transform.rotation;
       }
   }
   

Presumably the hideFlags ensure it isn't saved to the scene, and it doesn't even show up in the hierarchy. It does modify the scene though, even if you just select an object with this in it; definitely not ideal, but a small price to pay for actually having stuff placed on the ground. Anyway, not the most elegant solution (and I suspect I'll think of a better way in the morning), but it seems to get the job done for now.

The shader used for the gizmos is not settable, and it doesn't perform a depth test, so you're SOL with built-in Gizmos. You'll have to create your own objects/scripts that run in the Editor. This is a good use for the "EditorOnly" tag.