Context. While we have ample evidence of stellar analogues to the solar activity cycle for chromospheric activity, very little is known about stellar coronal cycles and their possible similarities to the solar behavior. Aims. An ongoing X-ray monitoring program of solar-like stars with XMM-Newton is performed to investigate coronal activity cycles. Methods. We used X-ray observations of the nearby binaries 61 Cyg A/B (K5V and K7V) and alpha Cen A/B (G2V and K1V) to study the long-term evolution of magnetic activity in weakly to moderately active G + K dwarfs over nearly a decade. Specifically we searched for X-ray activity cycles and related coronal changes and compared them to the solar behavior. Results. For 61 Cyg A we find a regular coronal activity cycle analog to its 7.3 yr chromospheric cycle. The X-ray brightness variations are with a factor of three significantly lower than on the Sun, yet the changes of coronal properties resemble the solar behavior, with stronger variations occurring in the respective hotter plasma components. 61 Cyg B does not show a clear cyclic coronal trend so far, but the X-ray data match the more irregular chromospheric cycle. The two alpha Cen stars exhibit significant long-term X-ray variability. alpha Cen A shows indications for cyclic variability of an order of magnitude with a period of about 12-15 years; the alpha Cen B data suggest an X-ray cycle with an amplitude of about six to eight and a period of 8-9 years. The sample stars exhibit X-ray luminosities ranging between L-X less than or similar to 1 x 10(26)-3 x 10(27) erg s(-1) in the 0.2-2.0 keV band and have coronae dominated by cool plasma with variable average temperatures of around 1.0-2.5 MK. Conclusions. Coronal activity cycles are apparently a common phenomenon in older, slowly rotating G and K stars. The spectral changes of the coronal X-ray emission over the cycles are solar-like in all studied targets.