With the exception of Okamiden, Tsumuji, and Away Shuffle Dungeon, many aren't really getting the "Zelda-like" component of the game(s) request.
Zelda means Puzzles. Zelda means thinking, Zelda means good, and people are either suggesting great games that are very un-zelda-like, or zelda-like games that aren't good/worth the time.
1st to point out that Mario and Luigi, Avalon Code, Kingdom Hearts, Tales of Innocense, or any other RPG/Action RPGs on page 1 are JUST RPGs. They often have no puzzles. Just battle and go to the next room/find exit. Thats not to say that they aren't great to play, they're just terrible suggestions for anyone wanting to seek Zelda-like gameplay.
Games that are acceptable to call Zelda-likes:
Golden Sun Dark Dawn, while classified as RPG, the psynergy you learn throughout the game are incorporated into out-side-battle gameplay where they are used to solve puzzles to get to the next room or uncover a secret/item/etc... Unlike other RPG's like Tales or Chrono Trigger, where all you're doing is just exploring areas trying to find the next room with almost no thinking. Golden Sun is one of the best examples when it comes puzzle solving that requires (x) item to solve (y) context puzzle.
Okamiden, easily my Top 5 games, is like Zelda without the touch controls. Brush Skills you learn are used to get past/solve obstacles that obstruct your path. Unlike Golden Sun, there is only minor backtracking.
Final Fantasy Rings/Echoes, which is kind of stretching it. While it does require some thinking/puzzle work in every room you go into, theres none of that bzns where you got to get a dungeon's item to progress. From start to finish, you already have most of what you need to progress through any dungeon. If a room requires you to use thunder on one item and Blizzard on the other, no big deal, you already have those items when you started.
Tsumuji - Its a clone in Japanese. I mean you can still play it blindly if you wanted to. If story is not priority.
Away Shuffle Dungeon not quite zelda-like the others, but still a good way to kill time.
Dungeon Raiders, just not worth playing, even for free.
Games that have been suggested but totally miss "zelda" feel:
Ninja Gaiden - I mean come on, really?
Dragon Quest IX - Just another RPG
Chrono Trigger - Just another RPG
Mario and Luigi - -__-
Secret of Mana - Just another RPG thats not even on DS (GBA)
Kingdom Hearts - Action RPG, but totally not Zelda-like
Tales of - Action RPG, but totally not Zelda-like
Avalon Code - Action RPG, but totally not Zelda-like
Avatar - -_____-
Lunar Knights - Action RP- you get the idea.
Lufia - Action RP- you still get the idea.
Dragon Ball - Same general control scheme as DS Zeldas, without almost zero of the puzzles/thinking.
Grand Theft Auto - .......