For SQL Server 2008, the best and index friendly way is
DELETE from Table WHERE Date > CAST(GETDATE() as DATE);
For prior SQL Server versions, date maths will work faster than a convert to varchar. Even converting to varchar can give you the wrong result, because of regional settings.
DELETE from Table WHERE Date > DATEDIFF(d, 0, GETDATE());
Note: it is unnecessary to wrap the DATEDIFF with another DATEADD