We present MEMprop, the adoption of gradient-based learning to train fully memristive spiking neural networks (MSNNs). Our approach harnesses intrinsic device dynamics to trigger naturally arising voltage spikes. These spikes emitted by memristive dynamics are analog in nature, and thus fully differentiable, which eliminates the need for surrogate gradient methods that are prevalent in the spiking neural network (SNN) literature. Memristive neural networks typically either integrate memristors as synapses that map offline-trained networks, or otherwise rely on associative learning mechanisms to train networks of memristive neurons. We instead apply the backpropagation through time (BPTT) training algorithm directly on analog SPICE models of memristive neurons and synapses. Our implementation is fully memristive, in that synaptic weights and spiking neurons are both integrated on resistive RAM (RRAM) arrays without the need for additional circuits to implement spiking dynamics, e.g., analog-to-digital converters (ADCs) or thresholded comparators. As a result, higher-order electrophysical effects are fully exploited to use the state-driven dynamics of memristive neurons at run time. By moving towards non-approximate gradient-based learning, we obtain highly competitive accuracy amongst previously reported lightweight dense fully MSNNs on several benchmarks.