Zuigan-ji Temple
Matsushima's top Zen temple with over a thousand years of history, but not a top choice for relaxation: tickets are sold from vending machines and guides shouting into megaphones herd tour groups through the temple, which has been turning into a museum with everything of interest partitioned off and/or packaged in glass cases. The approach with its moss-covered Kannon statues is atmospheric though. Repairs on the main temple roof started in 2009 and are projected to last 5 or more years.
Kanran-tei Pavilion
Originally built in Kyoto by famed Shogun Toyotomi Hideyoshi, this was eventually moved to Matsushima by Date Tadamune in 1645 and is the largest Momoyama-style tea house in Japan. An excellent place to stop for a ¥200 cup of tea traditional Japanese sweets included and a view of the Matsushima coastline.
Fukuura Island
Connected to the mainland by a long bridge, the island is crisscrossed in all directions by paths small and large, paved and muddy, well-trod and overgrown. A circuit of the island won't take more than an hour and there are some very sparsely beautiful spots to be seen. An interesting anecdote, there is a local superstition that crossing the bridge with a girlfriend/boyfriend will cause a breakup.