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We propose a size-consistent Grüneisen-quasiharmonic approach (GQA) to calculate the lattice thermal conductivity $κ_l$ where the Grüneisen parameters that measure the degree of phonon anharmonicity are calculated directly using first-principles calculations. This is achieved by identifying and modifying two existing equations related to the Slack formulae for $κ_l$ that suffer from the size-inconsistency problem when dealing with non-monoatomic primitive cells (where the number of atoms in the primitive cell $n$ is greater than one). In conjunction with other thermal parameters such as the acoustic Debye temperature $θ_a$ that can also be obtained within the GQA, we predict $κ_l$ for a range of materials taken from the diamond, zincblende, rocksalt, and wurtzite compounds. The results are compared with that from the experiment and the quasiharmonic Debye model (QDM). We find that in general the prediction of $θ_a$ is rather consistent among the GQA, experiment, and QDM. However, while the QDM somewhat overestimates the Grüneisen parameters and hence underestimates $κ_l$ for most materials, the GQA predicts the experimental trends of Grüneisen parameters and $κ_l$ more closely. We expect the GQA with the modified Slack formulae could be used as an effective and practical predictor for $κ_l$, especially for crystals with large $n$.