"Thank you" in Tibetan, Cantonese, and Mandarin
As others have correctly pointed out, the Tibetan expression for 'thank you' is ཐུགས་རྗེ་ཆེ་ (Wylie: thugs rje che). The approximate phonetic pronunciation in the Lhasa dialect is thu-je-che (the 'gs' and 'r' being silent). Literally, ཐུགས་རྗེ་ (thu-je) means 'compassion' and ཆེ་ (che) comes from the word ཆེན་པོ་ (chen-po), which means 'great'. So when you are thanking someone, you are basically telling them that they have 'great compassion'.

Although it may sound like the Chinese phrase 多謝, the similar pronunciation is just a coincidence and the two expressions have nothing in common whatsoever.