We present an X-ray spectral analysis of the nearby double radio relic merging cluster Abell 3376 (z = 0.046), observed with the Suzaku XIS instrument. These deep (∼360 ks) observations cover the entire double relic region in the outskirts of the cluster. These diffuse radio structures are amongst the largest and arc-shaped relics observed in combination with large-scale X-ray shocks in a merging cluster. We confirm the presence of a stronger shock (ℳ_W = 2.8 ± 0.4) in the western direction at r ∼ 26′, derived from a temperature and surface brightness discontinuity across the radio relic. In the east, we detect a weaker shock (ℳ_E = 1.5 ± 0.1) at r ∼ 8′, possibly associated with the “notch” of the eastern relic, and a cold front at r ∼ 3′. Based on the shock speed calculated from the Mach numbers, we estimate that the dynamical age of the shock front is ∼0.6 Gyr after core passage, indicating that Abell 3376 is still an evolving merging cluster and that the merger is taking place close to the plane of the sky. These results are consistent with simulations and optical and weak lensing studies from the literature.