I will describe a new bipartite Euler system-type construction system for GSp_4 and its inner forms, based on the special cycles appearing in the Kudla program (for instance, Shimura curves on Siegel threefolds). This leads to new results towards the Bloch-Kato Conjectures in ranks 0 and 1 for four-dimensional Galois representations associated to Siegel modular forms of parallel weights (3,3). The key ingredients of the proof are the structure of the supersingular locus for special fibers of GSpin_5 Shimura varieties at primes of both good and bad reduction, along with the representation theory of theta lifts underlying Kudla's program.