Cleaning ThreadMesses, especially your own, is being a GoodWikiCitizen. Especially if you find yourself about to add comments to a mature ThreadMode discussion, first consider refactoring it. Just make certain the thread is mature - it's all played out and cool.
The traditional way to do this is to separate the old threads into a page of their own called <PageX>Discussion. That way anyone who feels particularly attached to some particular edit can still find it easily and ReFactor it wherever they think it fits better.
This is important because often ThreadMess occurs when contributors add content that is almost on-topic for a page, but subtly misses the point that the WikiNature is trying to make; that is, the ThesisAntithesisSynthesis can't happen because there is too much noise. A page may contain very good content that is still noise to that page. If one comes across misplaced content cluttering up a ThreadMode discussion it is one's duty to move the good stuff to where it can be useful.
Then leave a note to show what the removed material was and where its new home is located. Lousy content -- chatter, drivel, and other fluff -- can be moved to a Discussion page without any mention at all. If it had value, it'll show up again.
If the discussion is very long, or if it recurs, also consider PageVersioning.
Discussion moved, of course, to ThreadMessDiscussion.
And: ConvertThreadModeToDocumentMode, RefactorByCondensingQuestionAnswerPair, RefactorByExtractingToPage, RefactoringWikiPages, RefactorWhileRespectingSignatures, SummariesOnTopDiscussionBelow, UseFactsToComplain, PlainEnglish, GentleReductionism, ThesisAntithesisSynthesis, OneWikiStyle, DeletionDiscussion