Assume your drop downs are in the range A2:D2

Select the sheet where you want this to happen.
Right click on the sheet tab and select View Code.
Copy/paste the code below into the window that opens.

Option Explicit
Private Sub Worksheet_Change(ByVal Target As Range)
  Application.EnableEvents = False
   On Error GoTo sub_exit
   If Not Intersect(Target, Range("A2")) Is Nothing Then
   Range("B2:D2").ClearContents
   End If
   If Not Intersect(Target, Range("B2")) Is Nothing Then
   Range("C2:D2").ClearContents
   End If
   If Not Intersect(Target, Range("C2")) Is Nothing Then
   Range("D2").ClearContents
   End If
sub_exit:
   Application.EnableEvents = True
End Sub

Return to Excel - hit ALT Q or click the top "X" to close the window.

Try it out.

First, this may not work in xl97.

xl97 has a bug that under certain conditions, the worksheet_Change event won't
fire.

Debra Dalgleish explains it in more detail:
http://contextures.com/xlDataVal08.html#Change

Anyway, you could use code like this that fires each time you make a change to a
cell in that worksheet.

Option Explicit
Private Sub Worksheet_Change(ByVal Target As Range)

   Dim myAddresses As Variant
   Dim aCtr As Long  'Address Counter
   Dim nCtr As Long  'next counter
   
   'change this to match your range
   'A1 controls B1 which controls C1 which controls E1
   myAddresses = Array("a1", "B1", "c1", "e1")
   
   'one cell only
   If Target.Cells.Count > 1 Then
       Exit Sub
   End If
   
   On Error GoTo ErrHandler:
   For aCtr = LBound(myAddresses) To UBound(myAddresses)
       If Intersect(Target, Me.Range(myAddresses(aCtr))) Is Nothing Then
           'not in that range, do nothing
       Else
           Application.EnableEvents = False
           For nCtr = aCtr + 1 To UBound(myAddresses)
               Me.Range(myAddresses(nCtr)).ClearContents
           Next nCtr
       End If
   Next aCtr
   
ErrHandler:
   Application.EnableEvents = True

End Sub