Development Hell: TGM4. It's been over a year and there's been very little word of its release in months, culminating with announcement of cancellation. It seemed to have been Saved from Development Hell with TGM2015, which is TGM4 in all but name, but that also suffered the same problem.
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 downloadable content; as a result, proper ARS can only be unlocked if you have an XBL Gold account.
For the same reason, Mihara has been unable to get a fourth TGM game published, as the TTC would probably look at it, notice its various Guideline variations, and say "no."
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. Since then, most, if not all, videos of TGM clones have been put back up, with no further action from Arika or TTC.
Torikan: The time-based checkpoints that trigger a Non-Standard Game Over if their requirements are not met (for example, 3'25" in TAP's T.A. Death mode at level 500 and 7'00" in Ti's Master mode at level 500). The term comes from toriaezu kansuto, or "counter-stopped for the time being". More details can be found here.
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.
No Export for You: The entire series, pretty much. Most players outside of Japan use clones or emulators to play, and a few are lucky enough to live near an actual TGM machine, and even fewer are able to find and afford to own the actual hardware. Mihara, the director of the series, openly hating Western players for pirating the game so heavily (never mind that they have literally no other options to play it), doesn't help. When an American player got GM grade on a legitimate TGM 1 board and showed it off on Twitter, he replied that he's no better since he's not doing anything to stop piracy and cloning of his games.
Revival by Commercialization: Although the memetically popular "Invisible Tetris" video and the like have been floating around since 2006, the series hit a second popularity spike amongst Westerners when it was streamed on Awesome Games Done Quick in 2015. Having the game be demonstrated by and explained by Westerners certainly helped a lot.
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.
Viral Marketing: Beginning in 2015, Arika vice president Ichiro Mihara launched the official TGM account, and encouraged players to show their support for the series using the #tgm_series hashtag whenever discussing TGM and posting their accomplishments. Although it's nowhere near a trending topic in any region, it has seen regular use, with Mihara periodically retweeting tweets (especially accomplishments) that fall under the tag.
What Could Have Been: TGM 1 was going to have a US release, as hinted in Dummied Out English-language text and manual. For whatever reason, the game ended up never leaving Japan.