We introduce and study the noncommutative modular function spaces of measurable operators affiliated with a semifinite von Neumann algebra and show that they are complete with respect to their modular. We prove that these spaces satisfy the uniform Opial condition with respect to \widetilde{ρ}-a.e.-convergence for both the Luxemburg norm and the Amemiya norm. Moreover, these spaces have the uniform Kadec-Klee property with respect to \widetilde{ρ}-a.e.-convergence when they are equipped with the Luxemburg norm. The above geometric properties enable us to obtain some results in noncommutative Orlicz spaces.