How Do I Copy/Paste All New Rows From All Worksheets To New Worksheet?

You can use ADO.Recordset[^] and Range.CopyFromRecordset[^] method.

Here[^] you'll find the proper connection string to MS Excel file.

The command would looks like this:

SELECT *
FROM (
    SELECT Field1, Field2, ... FieldN
    FROM [Sheet1$]
    UNION ALL
    SELECT Field1, Field2, ... FieldN
    FROM [Sheet2$]
    UNION ALL
    SELECT Field1, Field2, ... FieldN
    FROM [Sheet3$]
) 
WHERE DateField =>#2014/03/28#

Note: Do not forget about MS Excel limitations[^]. You need to restrict the count of records to the count of rows in a worksheet.