We describe the design and performance of a high-repetition-rate single-frequency passively Q-switched Yb:YAG microlaser operating near 1030 nm. By using short cavity length, an intracavity Brewster polarizer, and an etalon output coupler, we are able to produce ~1-ns-long single-frequency pulses at repetition rates up to 19 kHz without shot-to-shot mode hopping. The laser's output spatial mode is TEM 00 and its pulse energy varies between 31 µJ and 47 µJ depending on repetition rate. Its peak optical-to-optical efficiency is 22%.