We have searched for CP violation in the decays D-0 -> (KSP0)-P-0 where P-0 denotes a neutral pseudoscalar meson that is either a pi(0), eta, or eta' using KEKB asymmetric- energy e(+)e(-) collision data corresponding to an integrated luminosity of 791 fb(-1) collected with the Belle detector. No evidence of significant CP violation is observed. We report the most precise CP asymmetry measurement in the decay D-0 -> K-S(0)pi(0) to date: A(CP)D(0) -> K-S(0)pi(0) = (-0.28 +/- 0.19 +/- 0.10)%. We also report the first measurements of CP asymmetries in the decays D-0 -> K-S(0) -> and D-0 -> K-S(0)eta': A(CP)(D0 -> KS0 eta) =(+0.54 +/- 0.51 +/- 0.16)% and A(CP)(D0 -> KS0 eta') = (+0.98 +/- 0.67 +/- 0.14)%, respectively.