In theory, it should be possible to play blindfolded using the piece sounds. Easier said than done, but I'll attempt to work out the details as much as possible here.
Piece Sounds
I: Very distinctive, longer than the other sounds.
L: 3 beeps, ascending.
O: Short, low, descending.
J: 3 beeps, descending.
T: High-low-high scale.
Z: Descending scale.
S: 3 beeps, mid-low-high.
Opening
The sound isn't played for the first piece, so you'll have to place it without knowing what it is. However, it's always one of I, L, J, or T. One possible opening move is to DAS right, stop, rotate counterclockwise, then drop. Then fill in the bottom 3 rows of columns 1 through 8, and listen for the line clear sound.
If there's no line clear, then the first piece was an I:
But if there is a line clear, even though you don't know whether the first piece was an L, J, or T, you do know the shape of the board:
