By the time of the Age of Exploration, Europe and East Asia had been trading via the Silk Road for more than a thousand years. The only reason the Spanish and Portuguese attempted to find a maritime route to Asia was because the Venetians controlled the overland route (thanks to Marco Polo). Trade along this route was conducted mostly by middlemen, who would carry goods for one leg of the journey before selling it to the next guy.
There were, however, famous Asian travelers. Rabban bar Sauma was a Turkic Nestorian Monk who traveled from Yuan (Mongol) China to Europe as an ambassador. Going the other way, there was Ibn Battuta, a Berber from Morocco who traveled a lot and eventually ended up in China.