I remember reading somewhere that there is a kabbalistic tradition to say the parts of davening that are in Aramaic (e.g. the targum in Uva Letziyon) quietly. What is the source for this?
The Rama in Orach Chaim siman 132 seif 1 says that in Uvah LeTzion (kedusha de'sidra) the parts of the kedusha that are translated into Aramaic should not be said aloud.
The Mishnah Berurah sk4 says that the Aramaic should be said as an individual since "everything that is in Aramaic should not be said berabim"
The perush Matok MiDvash explains the reason behind this halacha according to the Zohar (chelek 2 132b) which states that the shoresh (root) of Aramaic is from the sitra achah and we need to diminish the power of the sitra achah by means of reciting it as an individual.
As an aside: The Mishna Berurah concludes that if an individual is not praying with a tzibur they are allowed to say the Aramaic aloud. This is quoted from the Sha'arei Teshuva. However according to other explanations for the Rama's halacha (some brought in the Peri Megadim) you might not be allowed to say the Aramaic aloud even when not praying with a tzibur
The Gemara in Shabbos 12b mentions that one shouldn't daven in Aramaic because angels don't understand Aramaic. The angels "carry" the prayers to G-d, and if they don't understand them, they won't take them because they don't want to take unacceptable prayers. However, in prayers which are a davar sheb'kedusha, the prayers are listened to directly by Hashem, and therefore Kaddish is said aloud in Aramaic (along the lines of the gemara there, that davening in the presence of the sick person can be done in Aramaic because Hashem is with the sick).