Is it permissible to pray three rak'ahs as a thank-you to Allah after each obligatory prayer, or is this considered an innovation (bid'ah)?

There is no 'gratitude prayer' (salat al-shukr) in Islamic law. Rather, what is prescribed is the prostration of gratitude (sujud al-shukr) when its cause is present. It is appropriate for one performing voluntary prayers to pray two rak'ahs at a time, based on the Prophet's (peace be upon him) saying: "The night prayer is two by two," meaning voluntary prayers are performed as two rak'ahs, not three. This is the view of the majority of scholars.