Monolithic integration of absorber layer and readout electronics is expected to greatly improve spatial resolution and sensitivity of X-ray imaging detectors. It requires, however, heteroepitaxial growth of thick, lattice, and thermally mismatched absorber layers on a Si substrate. Wafer bowing and layer cracks induced by temperature changes have so far appeared to be insurmountable obstacles in the way of realizing such a device. Here we present first results on a detector concept which does not suffer from such shortcomings. The absorber consists of closely spaced, tall Ge crystals, typically a few microns in width, each forming a heterojunction diode with the Si substrate. Electrical measurements on such diodes reveal reverse dark currents of the order of 1 mA/cm2, low enough for detector fabrication. We present a preliminary version of such a detector, where the pixel size is determined by the CMOS circuits rather than individual Ge crystals.