Something other solutions and guides for this game don't seem to mention is that this is the only achievement that has to be done in one run. All the other achievements can be mixed around and played in whatever order you choose, but this one has to be all levels in one go. At least it was that way for me; I did all the levels out of order, some of them several times, and the achievement didn't pop until I played all of them in order, by hitting Continue after finishing each level.
You can, however, screw up like I did and accidently hit RB in level 4 and still get the achievement if you pause and hit "restart level" and then beat the rest of the levels.
Also, you can use all powerups except the X button one. I used several shields and a few of the Y button ones in my successful run. I never touched B just in case. Basically, though, this achievement is for beating the game without ever touching LT, X or RB.