The Mira variable R Hydrae is well known for its declining period, which Wood & Zarro (1981) attributed to a possible recent thermal pulse. Here we investigate the long-term period evolution, covering 340 years, going back to its discovery in AD 1662. Wavelets are used to determine both the period and semi-amplitude. We show that the period decreased linearly between 1770 and 1950; since 1950 the period has stabilized at 385 days. The semi-amplitude closely follows the period evolution. Detailed analysis of the oldest data shows that before 1770 the period was about 495 days. We find no evidence for an increasing period during this time as found by Wood & Zarro. IRAS data shows that the mass loss dropped dramatically around AD 1750. The decline agrees with the mass-loss formalism from Vassiliadis & Wood, but is much larger than predicted by the Bloecker mass-loss law. An outer detached IRAS shell suggests that R Hya has experienced such mass-loss interruptions before. The period evolution can be explained by a thermal pulse occuring around AD 1600, or by an non-linear instability leading to an internal relaxation of the stellar structure. The elapsed time between the mass-loss decline giving rise to the outer detached shell, and the recent event, of approximately 5000 yr suggests that only one of these events could be due to a thermal pulse. Further monitoring of R Hya is recommended, as both models make strong predictions for the future period evolution. R Hya-type events, on time scales of 10^2-10^3 yr, could provide part of the explanation for the rings seen around some AGB and post-AGB stars. Comment: 13 pages. MNRAS, accepted for publication