On a 1d chain, if the atoms can move in all three directions, then yes, you would still have the factor of 3 out front. However, if the atoms were constrained to move only on the one-dimensional line, then it would be a factor of 1 out front.
If there is a factor of 3 out front (i.e., if the atoms can move in all three directions), then there are 3 acoustic modes --- two transverse and one longitudinal (this is true even in 1d). The speed of sound of the two transverse modes is the same by symmetry, but generally different from the speed of sound for the longitudinal mode (typically the speed of sound is higher for teh longitudinal mode).
The story is similar in 3d. Three acoustic modes. Two transverse (same velocity) and one longitudinal (higher velocity).
In the 2009 paper they state that motion perpendicular to the plane is to be ignored. So atoms move in 2 directions and you should expect a factor of 2 out front of the debye integrals.. and two acoustic modes -- one longitudinal and one transverse.