Suspensions of motile cells are model systems for understanding the unique mechanical properties of living materials which often consist of ensembles of self-propelled particles. We present here a quantitative comparison of theory against experiment for the rheology of such suspensions in extensional flows. The influence of motility on viscosities of cell suspensions is studied using a novel acoustically-driven microfluidic capillary-breakup extensional rheometer. Motility increases the extensional viscosity of suspensions of algal pullers, but decreases it in the case of bacterial or sperm pushers. A recent model [Saintillan, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2010, 81, 56307] for dilute active suspensions is extended to obtain predictions for higher concentrations, after independently obtaining parameters such as swimming speeds and diffusivities. We show that details of body and flagellar shape can significantly determine macroscale rheological behaviour.