It works, all right!

Let me tell you all how it's done:
I) Download either
GenuineCheck.exe ver. 1.7.36.0 or
GenuineCheck.exe ver. 1.7.69.2, as per
post #5, and keep it in a convenient directory. It requires no installation. But it does require a registered msxml3.dll (in doubt, refer to post #5).
II) Navigate to
Validate Windows Using an Alternate Method, as per
post #7, and click "Validate Now". It'll offer the download of "legitcheck.hta". Accept it and direct the download of this file to C:\WINDOWS\SYSTEM (or whatever the route to the SYSTEM directory in your machine is). Use the other link provided in post #7, in case you're interested in validating Office.
III) Go to a Microsoft download page that requires validation and click on the "Continue" button that is beside the "Validation Required" message on the orange ribbon. A box will open and offer to install "Windows Genuine Advantage" and you'll click on the "No" button. Then scroll down the page and click on the link "Genuine Windows Validation, Alternate Method".
IV) You'll arrive on a page that has two items: (1) Run the validation tool and (2) Enter your validation code.
Do not use the first item because, if you do it, it will download GenuineCheck.exe ver. 1.9.0009.1 or greater, which does not run in Win 9x/ME. I repeat, don't click on "Continue" below the item (1), or select Cancel, in case you've already done it.
V) Run GenuineCheck.exe ver. 1.7.36.0 or 1.7.69.2 instead, from the directory you'd saved it to, and use the "Copy to Clipboard" button in it, then close it. If it's the first time you run it, it'll complain it's outdated. Just run it once more, and it'll work.
VI) Go back to your navigator and paste the code you just obtained to the approppriate window below the item (2) in the page you were, and then click on the "Validate" button.
VII) You'll be automagically taken back to the download page you had originally navigated to in item III of this post, but now, instead of the orange ribbon, there will be a blue one, saying Genuine Microsoft Software, with a "Download" button beside it. Use that button to download normally.
Enjoy!
P.S.:
(i) Thanks are due to Marius '95, who originally reported (
link) that GenuineCheck.exe ver. 1.7.36.0 runs OK under Win 9x/ME. He's the real inventor of this procedure, although he did not describe it in so many words.
You do rock, Marius '95!

:
(ii) Thanks are due to jds, for
pointing out that GenuineCheck.exe ver. 1.7.69.2 also runs OK under 9x/ME.
You do rock too, jds!

(iii) Perhaps it's possible to simplify this procedure, by omitting step II above. I'm not positive that "legitcheck.hta" is really needed. If anyone would be willing to test this hypothesis and report, please do so!
(iv) And thanks also to loblo, for the info on the next post. Yes, loblo rocks too!