Transient rheological measurements are reported for a model polymeric bicontinuous microemulsion. The sample consists of a ternary blend of poly(ethyl ethylene) (PEE) and poly(dimethyl siloxane) (PDMS) homopolymers and a symmetric PEE-PDMS diblock copolymer. Steady-flow rheological data, reported previously, show four regimes as a function of increasing shear rate. Newtonian behavior is observed in regime I, followed by shear thinning in regime II. Flow-induced phase separation is the hallmark of regime III. The microemulsion starts ejecting homopolymer-rich phases, and the shear stress is independent of shear rate. In regime IV, complete phase separation occurs and the sample behaves like an immiscible blend. Transient rheological data on flow inception reveal linear viscoelastic response in regime I, and development of a stress overshoot in regime II. In regime III, a strong stress overshoot is observed, followed by a "shoulder" and a slow decay to the steady, rate-independent value. The normal stress shows a similar response as the shear stress. The transient morphology development has been further characterized using light scattering and microscopy. A "double-streak" pattern develops in light scattering on flow inception, which ultimately evolves into a "single-streak" pattern. Optical microscopy shows development of a string-like morphology, indicating a transition in lengthscale from tens of nanometers to microns. Transient rheological measurements on step changes in shear rate, both within and between regimes III and IV, and also on flow cessation, support the proposed morphologies.