Executive Meddling: Tetris: The Grand Master ACE. The Tetris Guideline, a series of rules that games carrying the "Tetris" name must abide by, forced significant changes to the gameplay mechanics. On top of that, Microsoft wanted Arika to hurry the game to make it a launch title for the Xbox 360 and incorporate downloadble content; as a result, proper ARS can only be unlocked if you have an XBL Gold account.
Arika has, on two occasions, had videos of TGM clones (such as Heboris and Texmaster) removed from YouTube. In the case of the second wave of removals, Arika denied doing so of their own volition, claiming that The Tetris Company made them do it, which leads to some Unfortunate Implications about TTC's possible intentions. Since then, most, if not all, videos of TGM clones have been put back up, with no further action from Arika or TTC.
The MOve Reset: TGM4:TMOR, for Arika's decision to make ARS an easier, "move reset" rotation system.
Keep Circulating the Tapes: Good luck trying to get any of the games legitimately. None of the TGM games have had console ports, forcing players to resort to purchasing arcade boards (not very cheap, especially when $100 for TGM1 is considered fair), visit an arcade with TGM (extremely rare outside of Japan; there is exactly one arcade in the entire United States with a TGM cabinet), or pirate. TGM ACE is more of a Guideline-compliant spinoff of TGM than a proper TGM game.
Screwed by the Network: It's widely speculated that TGM4 was cancelled because it was to be released alongside Tetris Giant and SEGA, the publisher, didn't want to release two Tetris games in one year.