The elastic properties and their pressure dependence of four B1-type alkaline earth oxides, MgO, CaO, SrO, and BaO, are calculated using the ab initio full-potential linear muffin-tin-orbital (FP-LMTO) generalized gradient approximated (GGA) method to elucidate their systematics. The calculated results agree quite well with the comparable experimental data. The large pressure dependence of c(11) and c(44) of MgO observed over 25 GPa is not predicted as well as the previous local-density approximation (LDA) calculations. It is inferred that the high-pressure measurement of elastic constant is quite sensitive to nonhydrostaticity. The deviation from the Cauchy relation and the elastic anisotropy are investigated. It is found that the interatomic interaction in SrO is nearest to the two-body force and the many-body contribution is largest in MgO. The elastic anisotropy in SrO and BaO are almost the same and only MgO has a large positive anisotropy under low pressure. The normalized elastic constants c(ij)' are introduced for more detail analysis. It is found that while there is no substantial difference in c(11)' and c(12)' of the four oxides, only c(44)' and its pressure dependence vary systematically. The difference in the Cauchy deviation and elastic anisotropy of alkaline earth oxides under pressure is characterized by c(44).