First of all, to clarify what the question is/should be, there's no obligation to wear tzitzis under normal circumstances, unless one is wearing a four cornered garment. Hence, the fact that tzitzis aren't in the list of clothing is not a problem; they aren't commanded in wearing an extra garment to put on tzitzis, just as no Jew is obligated (strictly speaking) to do so.
However, the kohanim actually should be obligated in tzitzis, because they might have been wearing a four-cornered garment: the me'il. However, this depends on what the me'il actually looked like (there are four different opinions). The real question is, if the Meil had four corners, did the kohanim put tzitzis on the me'il, and if not, why not?
The Radvaz in his commentary to the Rambam (Hilchos Klei Hamikdash 9:3) asks this question on the Rambam, and answers that despite the fact that the garment technically had four corners, these were all attached towards the top, and so the meil wasn't really a four cornered garment (this has significant halakhic importance).
The Minchas Chinuch (Mitzvah 99, by R. Yosef Babad) writes that the garments of the kohanim didn't require tzitzis because they belonged to hekdesh, and they weren't the property of the kohanim (a borrowed garment doesn't require tzitzis). After all, the Gemara in Kiddushin 54a states: that a kohen cannot use his garment to affect kiddushin, which requires ownership of the object used for betrothal. However, the question of whether or not kohanim own their garments isn't necessarily so clear-cut.
R. Gershon Chanoch Henoch Leiner, in his book "Ein HaTecheilis" argues on the Minchas Chinuch and explains that the reason for the exemption from tzitzis is merely a 'gezairas hakasuv'. In other words, Hashem told Moshe to make a Meil, and the description doesn't include tzitzis, end of story.
He also quotes a Zohar that explains this 'geziras hakasuv' by saying that the bells and pomegranates that were on the bottom of the Meil served the same purpose as the tzitzis serve on a regular garment.
(All of these opinions are quoted and discussed by R. Menachem Mendel Kasher in Torah Shelaima, vol. 23 pg 177)