Two-dimensional crystals such as graphene and transition-metal dichalcogenides demonstrate a range of unique and complementary optoelectronic properties. Assembling different two-dimensional materials in vertical heterostructures enables the combination of these properties in one device, thus creating multifunctional optoelectronic systems with superior performance. Here, we demonstrate that graphene/WSe2/graphene heterostructures ally the high photodetection efficiency of transition-metal dichalcogenides with a picosecond photoresponse comparable to that of graphene, thereby optimizing both speed and efficiency in a single photodetector. We follow the extraction of photoexcited carriers in these devices using time-resolved photocurrent measurements and demonstrate a photoresponse time as short as 5.5 ps, which we tune by applying a bias and by varying the transition-metal dichalcogenide layer thickness. Our study provides direct insight into the physical processes governing the detection speed and quantum efficiency of these van der Waals heterostuctures, such as out-of-plane carrier drift and recombination. The observation and understanding of ultrafast and efficient photodetection demonstrate the potential of hybrid transition-metal dichalcogenide-based heterostructures as a platform for future optoelectronic devices.