In this paper, a practical millimeter-wave (mmWave) channel estimation technique for intelligent reflecting surface (IRS)-assisted single-user multiple-input multiple-output (SU-MIMO) systems is proposed. The proposed technique utilizes both uplink and downlink signalings as well as analog feedback. While the typical estimation techniques require high training overhead, the proposed technique employs an extremely short training sequence length. It is verified through the simulations that the proposed estimation technique can achieve a higher effective spectral efficiency due to its low training overhead than that of existing estimation methods.