In Russian pronunciation the second "t" is palatalized and the vowel following it is a diphthong /ja/. The "t"s don't have any aspiration. /tat'jana/ It transforms the sound-form completely. It sounds harsher in Russian.
Another pronunciation is: Tah-tee-anna Also it's interesting to add that there's a version of the name "Tatiana", of Finnish origin, named "Taina" which originated the name "Tainara" (Tah-ee-nah-rah).
I am a Canadian with a daughter named Tatiana. I still love the name; the only issue has been pronunciation. In the beginning we tried to get people to pronounce it the correct, Russian way: Tat-ya-na. We gave up because most people here say it Ta-tee-ann-ah. She now uses Tiana as a nickname part of the time.