The kendama actually entered Japan from the West in the 18th century. Originally used as part of a drinking game in the Edo Period (1603-1868), it grew in general popularity over time, especially with kids. You can hold the kendama in different ways, but your goal is (generally) to land the ball on any of the three cups or on the spike at the top. This toy has had a recent resurgence, and you can see tournaments not only in Japan, but around the world!
The humble yo-yo traces its origins back to ancient Greece and possibly even ancient China. They've surged in and out of popularity in various places around the world at various times, and were even popular in Europe in the late 18th century before achieving their most significant resurgence thanks to Pedro Flores, who brought them—and the name yo-yo—to the U.S. from the Philippines in the late 1920s.
In Japan, the yo-yo was very popular during the Edo Period, having arrived from China. They've come in and out of fashion several times since, and it's not uncommon to spot yo-yos—or references to them—in anime and manga. There are countless Japanese websites that will teach you yo-yo tricks, and the World Yo-Yo Contest 2015 was even held in Tokyo. The yo-yo is definitely a part of Japanese culture!
A temari is a very traditional toy that was imported from China around the seventh century. Temari are handballs made from parts of old kimono. Throwing a temari came to be a particularly popular girls' game at New Year's during the Edo and Meiji (1868-1912) Periods, though before and thereafter it was played by pretty much anyone year-round.
This ball-kicking game was introduced to Japan from China around the year 600. The ball (mari) is made of deerskin, and players may use any body part except their arms and hands to keep the ball aloft. Nowadays, it's still played at some Shinto shrines for festivals, most famously at Kyoto's Shimogamo Shrine in early January. To mark the occasion, the players all dress in kariginu, traditional robes of the Asuka Period (592-710). This is definitely a very interesting game to play and to watch!
While these toys are old, they still exist in the present day. If you're into classic Japanese culture, take a minute to check them out!