My list of recommended Puzzle-theme games that I've played in no particular order (that do not include Umihara Kawase)
1.
Series: Panel de Pon/Puzzle League
Most Recommended Title: Panel de Pon (Super Famicom/Wii U Virtual Console (JP))/Tetris Attack (SNES)
Latest in the series: A Little Bit of… Puzzle League (Europe/Oceania)/Puzzle League Express (US), (Nintendo Dsiware/Nintendo 3DS eShop)
Vague Description: Developed by Intelligent Systems, the original title is Panel de Pon for the Super Famicom and when it was released outside Japan, it was palette swapped replacing the character Lip and others with Yoshi and others and released as Tetris Attack. Future releases outside of Japan not called Panel de Pon had to be titled Puzzle League because Nintendo would have to pay licensing for the Tetris Brand. This is one of the reasons Tetris Attack is not available on the Virtual Console. If anything I'd love for them to then translate and localise the original Panel de Pon and release it on the Virtual Console outside of Japan then, it shouldn't even take long or cost much because the game doesn't even have that much text to translate.
If I remember correctly, Ubisoft has the licensing to Tetris now which is why Tetris for the Gameboy was removed from the Virtual Console 3DS service and Tetris Axis was removed from the Nintendo 3DS eShop. (And we all know how Ubisoft's version of Tetris turned out. Hah!
https://www.youtube.com/watch?v=NvHIpxJE2Ss (Video contains swearing))
The latest game in the Puzzle League series is a single player version of the game and is the equivalent to Tetris as in, it's just the puzzles and no mascot/characters like the other iterations of the games in the series. If you cannot get Panel de Pon, this one is still playable and you can get it on your Nintendo 3DS off of the eShop. Since it is a DsiWare game it is saved to your system memory and not the SD card.
2.
Series: Magical Drop
Most Recommended Title: Magical Drop III [Preferably the Japanese version] (Arcade, Neo Geo, Neo Geo CD, Sega Saturn, PlayStation, Zeebo, Game Boy Color (titled as Magical Drop), Wii Virtual Console)
Latest in the series: Magical Drop V (Steam)
Vague Description: Developed by Data East (Which went bankrupt in 2003), most of Data East IPs are owned now by G-Mode which is mostly known Japanese Mobile Games.
You can get Magical Drop III off of the Wii Virtual Console, unfortunately assuming you don't have a Japanese Wii, you will end up playing the English version of the game which removes the Japanese Voice Dubbing for each individual character with some crappy generic general voice dub which is the same for all characters, the story mode is butchered with a removed difficulty level, I think I already explained enough. If it is possible for you, play the Japanese version since it is the definitive version, only with the English version with regards to gameplay you just lose the hardest difficulty level in story mode.
In the latest game entry, Magical Drop V
http://store.steampowered.com/app/204960/ was developed by Golgoth Studio (which have no relation to past Magical Drop games), the game was apparently a buggy mess at launch. I remember getting the game for $5 sometime after launch and now the game is 99 cents! and that is not a sale price... according to steamspy it has only sold around 30,000 copies. (For what is now a 99 cent game!)
It is still a cheap entry into the series, they apparently did fix most game-breaking bugs however, the game lacks a Puzzle Mode so it only contains Story mode and Multiplayer.
What makes things worse is that the character selection is smaller than in Magical Drop III, which affects character attacks. It may not matter to you if you've never played the series although it feels like losing characters in a fighting game.
There is also a demo for Magical Drop V on steam if you don't want to waste 99 cents although you could still get a refund anyway thankfully now.
I have no clue if G-Mode would ever consider porting their old games to Steam, they didn't even publish Magical Drop V.
3.
Series: Catherine
Most Recommended Title: Catherine (Xbox 360/PS3)
Latest in the series: Catherine (Xbox 360/PS3)
Vague Description: Developed by Atlus, I would describe it as a puzzle-platformer. You go about moving blocks mainly creating stairs because it is literally a race to the top to escape. It has a pretty good story to go with it involving relationships.
There hasn't been a sequel to this game, Atlus is currently owned by Sega. I don't know if Sega would ever make Atlus put some of their games on Steam. It surprises me how Atlus aren't interested in PC considering their Shin Megami Tensei series always involves computer programming/hacking in detail.
4.
Series: Dr. Kawashima's Brain Training: How Old Is Your Brain?/Brain Age: Train Your Brain in Minutes a Day!
Most Recommended Title: (Undecided)
Latest in the series: Brain Age: Concentration Training (US)/Dr Kawashima's Devilish Brain Training: Can You Stay Focused? (Europe/Oceania - Unreleased)
Vague Description: I seriously don't get why they haven't released Devilish Brain Training in Europe/Oceania, it was suppose to come out in 2013 and apparently every investors meeting to now you will see the game still TBD (to be dated) for release in their release list of games. Are you kidding me?
https://www.nintendo.co.jp/ir/pdf/2016/160202_2e.pdf <-- go to Page 6 and look in the lower left box highlighting europe, this file is from January 2016 for their 3rd quarter Fiscal year 2015(2016?).
5.
Series: Picross
Most Recommended Title: (Undecided)
Latest in the series: Pokemon Picross (Nintendo 3DS eShop)
Vague Description: The free-to-play game of Pokemon Picross was my first venture into picross and it is interesting. It's something akin to sudoku or minesweeper where you mark boxes on a grid to reveal a picture. There are tutorials I think for every game I've listed on this list. I think the picross series started from the Super Famicom era and has persisted since then. It is a Nintendo-developed game.
6.
Series: Picross 3D
Most Recommended Title: (Undecided)
Latest in the series: Picross 3D 2 (Soon to be released outside of Japan)
Vague Description: A 3D version which adds a new level of play, instead of a picture, it becomes an object. So it may help you figure out what it ends up being.
7.
Series: Zero Escape
Most Recommended Title: Zero Escape: Nine Hours, Nine Persons, Nine Doors (Nintendo DS)
Latest in the series: Zero Escape: Virtue's Last Reward (Nintendo 3DS/PS Vita)
Vague Description: Developed by Chunsoft and Directed by Kotaro Uchikoshi, Zero Escape: Nine Hours, Nine Persons, Nine Doors is a Visual Novel game with puzzle elements. I highly recommend the game because of the story, although yes, you do puzzles to progress through the game. I also recommend that you play in sequential order, first play Zero Escape: Nine Hours, Nine Persons, Nine Doors and then its sequel Zero Escape: Virtue's Last Reward.
It's also the kind of game you'd have to play without getting spoiled.
The third game in the series will be coming out this year for Nintendo 3DS/PS Vita in the US, I do not know what the European/Oceania release will be.
https://www.youtube.com/watch?v=sITddSPNB48 <-- (This is a reaction to the third game being announced at a convention. There are no story related spoilers. I'm guessing you should avoid reading the comments section in case there are spoilers.)
8.
Series: Ghost Trick
Most Recommended Title: Ghost Trick: Phantom Detective (Nintendo DS/iOS)
Latest in the series: Ghost Trick
Vague Description: Developed by Capcom and Directed by Shu Takumi who is known for creating the Ace Attorney series, you control a ghost that can possess objects which you use to solve puzzles. The story is good so once again you'd have to avoid spoilers.
9.
Series: Boxboy!
Most Recommended Title: Boxboy! (Nintendo 3DS eShop)
Latest in the series: Boxboy! One More Box (Nintendo 3DS eShop - Japan Only (Currently))
Vague Description: Developed by HAL Laboratory, it is a puzzle-platformer. You move a character that is box-shaped whom can make more boxes appear out of himself which you can use to solve platforming puzzles to get to the end of a stage. It's simple but is a good game.
10.
Series: Meteos
Most Recommended Title: Meteos (Nintendo DS)
Latest in the series: Meteos Wars (Xbox 360)
Vague Description: Developed by Q Entertainment and Sora Ltd. where Masahiro Sakurai (known for Super Smash Bros.) was a designer on the original game . You have blocks falling down into your planet and you match three or more to have them rocket propel back out into space. Apparently the IP for Meteos and Lumines(Not related to Meteos but is developed by Q Entertainment) were acquired by Japanese smartphone developed Mobcast.
11.
Series: Pullblox/Pushmo
Most Recommended Title: (Undecided)
Latest in the series: Stretchmo/Fullblox (Nintendo 3DS eShop)
Vague Description: Developed by Intelligent Systems, in the original game, it is a platformer-puzzle game where you pull blocks to reach a flag funnily enough its not the same as Catherine just going by this description. The blocks are multi-layered and represent some kind of object like an animal or such. Later iterations of the games have new mechanics such a falling blocks and stretching blocks(?).
Oh that took long enough, if you have any questions feel free to ask.