The ATLAS experiment at the LHC has measured the centrality dependence of charged particle pseudorapidity distributions over vertical bar eta vertical bar < 2 in lead-lead collisions at a nucleon-nucleon centre-of-mass energy of root s(NN) = 2.76 TeV. In order to include particles with transverse momentum as low as 30 MeV, the data were recorded with the central solenoid magnet off. Charged particles were reconstructed with two algorithms (2-point tracklets and full tracks) using information from the pixel detector only. The lead-lead collision centrality was characterized by the total transverse energy in the forward calorimeter in the range 3.2 < vertical bar eta vertical bar< 4.9. Measurements are presented of the per-event charged particle pseudorapidity distribution, dN(ch)/d eta, and the average charged particle multiplicity in the pseudorapidity interval vertical bar eta vertical bar < 0.5 in several intervals of collision centrality. The results are compared to previous mid-rapidity measurements at the LHC and RHIC. The variation of the mid-rapidity charged particle yield per colliding nucleon pair with the number of participants is consistent with lower root s(NN) results. The shape of the dN(ch)/d eta distribution is found to be independent of centrality within the systematic uncertainties of the measurement.