We report on the results of a four-year long X-ray monitoring campaign of the central 1.2 square degrees of our Galaxy, performed with Chandra and XMM-Newton between 2005 and 2008. Our study focuses on the properties of transient X-ray sources that reach 2-10 keV luminosities of L X 10 34 erg s −1 for an assumed distance of 8 kpc. There are 17 known X-ray transients within the field of view of our campaign, eight of which were detected in outburst during our observations: the transient neutron star low-mass X-ray binaries GRS 1741-2853, AX J1745.6-2901, SAX J1747.0-2853, KS 1741-293 (all four are also known X-ray bursters), and GRO J1744-28 (a 2.1 Hz X-ray pulsar), and the unclassified X-ray transients XMM J174457-2850.3, CXOGC J174535.5-290124 and CXOGC J174541.0-290014. We present their X-ray spectra and flux evolution during our campaign, and discuss our results in light of their historic activity. Our main results include the detection of two thermonuclear X-ray bursts from SAX J1747.0-2853 that were separated by an unusually short time interval of 3.8 min. Investigation of the lightcurves of AX J1745.6-2901 revealed one thermonuclear X-ray burst and a ∼1600-s long X-ray eclipse. We found that both XMM J174457-2850.3 and GRO J1744-28 displayed weak X-ray activity above their quiescent levels at L X ∼ 10 33−34 erg s −1 , which is indicative of low-level accretion. We compare this kind of activity with the behaviour of low-luminosity X-ray transients that display 2-10 keV peak luminosities of L X ∼ 10 34 erg s −1 and have never been seen to become brighter. In addition to the eight known X-ray transients, we discovered a previously unknown X-ray source that we designate XMMU J174654.1-291542. This object emits most of its photons above 2 keV and appears to be persistent at a luminosity of L X ∼ 10 34 erg s −1 , although it exhibits strong spectral variability on a time scale of months. Based on its X-ray properties and the possible association with an infrared source, we tentatively classify this object as a cataclysmic variable. No new transients were found during our campaign, reinforcing the conclusion of previous authors that most X-ray transients recurring on a time scale of less than a decade have now been identified near the Galactic centre.