Psychiatry rhymes with:
altimetry, asymmetry, banditry, basketry, bigotry, cabinetry, circuitry, dissymmetry, gadgetry, geometry, helotry, idolatry, optometry, poetry, punditry, puppetry, rocketry, spectrometry, symmetry, telemetry, toiletry, zealotry
Psychiatry sounds like:
pacesetter, pachter, paster, pasteur, pastor, pastora, pastore, pastry, pasture, pasztor, peachtree, pesatori, pescatore, pester, pfiester, pfiesteria, pfister, pfisterer, picketer, picture, pistor, pistorio, poster, posteraro, posterior, posture, postwar, pusateri