Whatever the belief system or whoever they're devoted to, shrines have a way of making you contemplative. Check out Iya Shrine and share a moment of silence with the locals.
You're spoiled for sightseeing choice when visiting Matsue. Iya Shrine is a great place to start here, but there are numerous other interesting sights to check out. Adachi Museum of Art, Yuushien Garden, and Yaegaki Shrine are near the sacred place and and will cap off your day nicely.