Magnetic properties of three isostructural compounds BaMP2O7 (M = Co, Ni, and Cu) were investigated by dc and ac magnetization and specific heat measurements. BaCuP2O7 was shown to be an excellent quasi-one-dimensional linear-chain Heisenberg antiferromagnet with an exchange constant (J/k(B)) of 103.8 K (Hamiltonian H = J Sigma SiSh1) and a temperature for the long-range magnetic order (T-N) of 0.81 K giving the ratio k(B) T-N/J = 0.78%. BaCoP2O7 and BaNiP2O7 exhibited long-range antiferromagnetic order at T-N = 10.4 and 10.1 K, respectively. BaCoP2O7 and BaNiP2O7 showed a large contribution of the short-range correlation above TN. BaNiP207 remained in the antiferromagnetic state up to 90 kOe at 2 K, whereas BaCoP2O7 demonstrated two metamagnetic phase transitions at about 52 and 71 kOe at 2 K if the magnetic field was parallel to the easy direction. BaMP207 melted incongruently at 1323 K (M = Co), 1344 K (M = Ni), and 1338 K (M = Cu).