We develop a new multiple input multiple output (MIMO) transceiver architecture – Differential Beamspace MIMO (DB-MIMO) for high-dimensional MIMO systems. DB- MIMO enables linear quasi-coherent interference suppression between multiple spatially multiplexed data streams in con- junction with differential transmission. The differential nature of DB-MIMO is particularly attractive in emerging high- frequency systems, e.g. centimeter-wave (cmW) and millimeter- wave (mmW) systems, where the requirement of a phase- coherent local oscillator at the receiver can be challenging. The beamspace approach is naturally relevant to tame the complexity of massive MIMO due to the expected channel sparsity in beamspace, especially at cmW and mmW. We focus on downlink multiuser systems in which an access point equipped with a high-dimensional antenna array serves multiple single- antenna mobile stations (MSs). First, the concept of dominant multi-beam selection is discussed that enables transceivers whose complexity tracks the number of MSs. Then the recently introduced concept of differential MIMO is applied in which a quasi-coherent channel estimate is obtained from differential measurements. This enables the development of low-complexity quasi-coherent DB-MIMO transceivers. Numerical results are presented to demonstrate the minimal performance loss of the low-complexity DB-MIMO transceivers in comparison with their full dimensional coherent counterparts.