Taptiles

To get this achievement you do need to play Dash Mode.
You need to build up your speed bonus and each time it reaches a multiple of 15 a pair of special tiles will appear.
When you get the Gem Mode special tiles and match them the game enters Gem Mode and you need to clear 3 puzzles before the Gem Mode ends in about 10 seconds.

To give yourself the best possible chance you should wait until there are only 4 tiles left, the 2 special Gem Mode tiles and 2 other tiles before matching the Gem Mode tiles to start Gem Mode. Then you just have a single match to make and thats 1 puzzle complete.
When the next puzzle comes up you need to tap away as fast as possible (I played on a touch screen). You should reach the next set of special tiles for your speed bonus very quickly and with any luck you will get the Wild Tile special tiles.
You need to leave the Wild Tiles as the last pair you match to complete the puzzle. If you do this then the next puzzle that comes up while still in Gem Mode will be completed automatically by the Wild Tiles.
Thats 3 puzzles complete in the one Gem Mode.

You do need a bit of luck for the Wild Tile special tiles to come up but since there are only a few types of special tiles this doesn't take many attempts to happen. It also works out easier if the second puzzle is one of the smaller ones.

I don't disagree with anything mentioned by kornfish27, but I did find another helpful way to complete this that at least worked for me somehow by luck.

During a dash run, one possible outcome is to actually get a second gem tile during the first gem tile. I didn't exactly get the Wild Tile but was still about to get the Gemologist Achievement during the second gem tile since I finished the 3rd round of Rush during the time granted by the second gem tile.

This is extremely rare probably, but it gives another possibility to win it if you're having trouble getting it based on the other solution.