it's considered a virtue that its intervals sound cleaner than equally tempered ones.

Another potential fix is to adjust timbres (i.e. sound spectra) so that they sound cleaner in equal temperament. See this example (MP3) from William Sethares' work (ironically, the only 12-TET piece from his freely-available samples). Sounds kind of uncanny and off-key to me, but that could be due to being unfamiliar with alt. tunings. YMMV.

ETA: The Hammond organ also used 12-TET frequencies to generate its "harmonic partials", so it was effectively just as "clean" in 12-TET as other instruments are in just intonation. On the other hand, many people would judge the effect as excessively "bland" and "indistinct". But the sound spectrum of the Hammond organ was not very complex to begin with; applying the same fix to other instruments will probably give more appealing results.