Combining Clarke's model with first-principles calculation of average sound velocity, the minimum lattice thermal conductivities (kappa(min)) of Y3Al5O12 (YAG), YAlO3 (YAP) and Y4Al2O9 (YAM) are predicted to be 1.59, 1.61, and 1.10 W.(m.K)(-1), respectively. The weak Y-O polyhedra provide "weak zones" that scattering phonons and lead to the low kappa(min) of ternary Y-Al-O compounds. In addition, the extremely low kappa(min) of YAM is attributed to its higher levels of local disorder of crystal structure and weaker chemical bonding compared with those of YAG and YAP. Inspired by theoretical predictions, dense and phase-pure YAM is synthesized and the experimental thermal conductivity is only 1.56 W.(m.K)(-1) at 1273 K. Finally, YAM is highlighted as a potential thermal barrier material for its low thermal conductivities at temperatures from 473 to 1273 K.