Extensive research has demonstrated the potential of cell viscoelastic properties as intrinsic indicators of cell state, functionality, and disease. For this, several microfluidic techniques have been developed to measure cell viscoelasticity with high-throughput. However, current microchannel designs introduce complex stress distributions on cells, leading to inaccuracies in determining the stress–strain relationship and, consequently, the viscoelastic properties. Here, we introduce a novel approach using hyperbolic microchannels that enable precise measurements under a constant extensional stress and offer a straightforward stress–strain relationship, while operating at a measurement rate of up to 100 cells per second. We quantified the stresses acting in the channels using mechanical calibration particles made from polyacrylamide (PAAm) and found that the measurement buffer, a solution of methyl cellulose and phosphate buffered saline, shows strain-thickening following a power law up to 200 s⁻¹. By measuring oil droplets with varying viscosities, we successfully detected changes in the relaxation times of the droplets and our approach could be used to get the interfacial tension and viscosity of liquid–liquid droplet systems from the same measurement. We further applied this methodology to PAAm microgel beads, demonstrating the accurate recovery of Young's moduli and the near-ideal elastic behavior of the beads. To explore the influence of altered cell viscoelasticity, we treated HL60 human leukemia cells with latrunculin B and nocodazole, resulting in clear changes in cell stiffness while relaxation times were only minimally affected. In conclusion, our approach offers a streamlined and time-efficient solution for assessing the viscoelastic properties of large cell populations and other microscale soft particles.