I solved this myself, but with a different workaround. Although I have to admit I don;t understand why it works.
The cells I was talking about were changing as a result of a lookup that was referencing values that changed when a refresh was run (from a web query). In case anyone's interested, this is what I used...
At the end of the refresh sub, I call this one:
'Inserts the date the cells were last updated (the date the refresh ran)
ActiveCell.FormulaR1C1 = "=NOW()"
Selection.PasteSpecial Paste:=xlValues, Operation:=xlNone, SkipBlanks:= _
Application.CutCopyMode = False