Whatever the faith or whoever they're dedicated to, shrines have a way of making you reflect. Check out Takeuchi Shrine and share a silent moment with the local faithful.
You'll be spoiled for sightseeing choice when in Matsue. Takeuchi Shrine is a good place to start here, but there are plenty of other fine sights to consider. Adachi Museum of Art, Matsue Castle, and Tamatsukuri Onsen are not far from the sacred place and will top off your day nicely.