Thanks very much Roy and Allen (again).  I modified what you suggested to do
the trick.  Just for completeness and to help anyone else who has this issue,
the final code is:

   Dim strSQL As String
   Dim tbl As TableDef
   
   ' Used for SQL Server database which attaches a "dbo_" to the front of
table names.
   strSQL = "SELECT * FROM MSysObjects WHERE Type = 4;"            ' Type 4
= ODBC
   
   If funRecordCount(strSQL) > 0 Then                              ' Check
if a record is found
       For Each tbl In CurrentDb.TableDefs
           If Len(tbl.Connect) > 0 Then
               tbl.Name = Replace(tbl.Name, "dbo_", "")
           End If
       Next
       Set tbl = Nothing
   End If

funRecordCount is a generic function I use in the application

Public Function funRecordCount(strSQL As String) As Integer
   Dim dbs As Database
   Dim rst As Recordset
   
   On Error GoTo Error_funRecordCount
   
   Set dbs = CurrentDb
   Set rst = dbs.OpenRecordset(strSQL, dbOpenSnapshot, dbSeeChanges)
   
   ' Find the number of records.  First test records were found.
   If (rst.EOF = True) And (rst.BOF = True) Then
       funRecordCount = 0
      Else
       rst.MoveLast
       funRecordCount = rst.RecordCount
   End If
   
Exit_funRecordCount:
   Set dbs = Nothing
   Set rst = Nothing
   Exit Function
   
Error_funRecordCount:
   MsgBox "Error in funRecordCount " & Err.Number & " - " & Err.Description
   GoTo Exit_funRecordCount
   
End Function