Of course. Ibn Battuta, Xuanzang, Fa Xian, to name a few. But do note: “Well-known” is relative. Ibn Battuta is, for example, rightly famous throughout the Muslim world, whereas Marco Polo is virtually unknown there.

#Ibn Battuta#

A Moroccan Muslim of the 14th century, Ibn Battuta departed his hometown of Tangiers (incidentally, in the same year Marco Polo died), to undertake the hajj, and ended up travelling not only to holy Makkah and Madinah, but also – over nearly three decades – across the length of the Maghrib; Mali; Al-Andalus; Egypt; West, Central, and South Asia; Sumatra; coastal East Africa; and China. Visiting such august metropolises as Cordoba, Alexandria, Damascus, Baghdad (though much-diminished), Constantinople, Bukhara, Samarqand, Delhi, Guangzhou, Hangzhou, and Dadu, he made a living serving as a qaḍi, and met with various sovereigns, some of whom bestowed gifts and appointments on him. In the end, he returned to his homeland, where he spent the remainder of his years, and dictated an account of his travels, with the flowery title تحفة النظار في غرائب الأمصار وعجائب الأسفار (A Masterpiece to Those Who Contemplate the Wonders of Cities and the Marvels of Travelling.)

While Ibn Battuta probably was the most widely-travelled man of his age, he was hardly the only one who covered great distances. Persian and Arab Muslim traders were long-established in the flourishing Indian Ocean trade, and there were Muslim (and some Jewish) enclaves from Zanzibar to Malabar to Malacca to Guangzhou. Many Chinese traders also settled in Coromandel and Malabar, especially when the Song rulers eased travel restrictions. But their records are naturally commercial. Furthermore, from all over the world, Muslims – who could – pilgrimaged to Makkah (though it would’ve scarcely been considered “exotic.”)

#Xuanzang#

Xuanzang was a Chinese Buddhist monk who, in the seventh century, journeyed to India via Turkestan and Afghanistan to procure Buddhist scriptures. This famous journey serves as the inspiration for the Ming fantasy novel Journey to the West. (Though I don’t recommend reading it; it’s really repetitive). At any rate, Xuanzang’s account is recorded in the 大唐西域記 (Great Tang Records on the Western Region), compiled on his return to China.

He was not the only one of course. Hundreds of Buddhist missionaries and other actors from India, China, Central Asia, and Persia had long wandered back and forth across these regions in the first millennium CE, acting as crucial agents in cultural and doctrinal exchange.

#Fa Xian#

About two centuries prior to Xuanzang, this Chinese monk (accompanied by others) went to India by the same route for the same reason. Visiting many sites of Buddhist pilgrimage, he returned to China after nearly two decades – via the maritime route.

As Fa Xian’s 佛國記 (Record of the Buddhist Kingdoms) was the first eyewitness account of Buddhist sites in Central and South Asia written in Chinese, it became very popular amongst the clergy there, developing their perception of India as a sophisticated civilization.

#Zheng He#

As one of the measures to legitimize his rule, Yongle, the third Ming emperor, appointed various military eunuchs to lead diplomatic missions to tributary states in Southeast Asia. Most famous of these is Zheng He, a Muslim eunuch, who led six such missions at the head of a grand fleet to places over the Indian Ocean: Malabar, the Persian Gulf, Arabia, and the East Africa.

Curiously, there is a popular impression that Zheng He was a sort of Asian Columbus. The evidence speaks clearly against this. Zheng He was an envoy charged with making states acknowledge Ming suzerainty; and was visiting lands long-known to traders, even if the Chinese were unfamiliar with them. Whereas Columbus’ was a commercial venture, that later acquired dimensions of conquest.

Detailed reports on the lands visited were prepared by Ma Huan, a Chinese translator, who accompanied Zheng He on three of his expeditions. The title of this compilation is 瀛涯勝覽 (General Survey of the Ocean’s Shores.)

#Malik Ambar#

From at least the fifteenth century, African slaves had been exported to India by Ethiopians, whose masters fed, housed, and educated them, receiving unswerving loyalty in return. These slaves served as domestic servants or as military soldiers and commanders.

One of these was Malik Ambar, who was captured and sold in the Red Sea markets, and eventually reached the Deccan, as part of a purchase by the peshwa of the sultanate of Ahmadnagar. He rose to become de facto ruler of the sultanate following the invasion of Mughal forces who occupied the capital. Using his Maratha and African warriors, Malik Ambar immensely frustrated the Emperor Jahangir’s campaign against Ahmadnagar with his skillful application of guerrilla tactics.

#Ibn Faḍlān#

Ibn Faḍlān was part of an Abbasid embassy that left Baghdad, eventually reaching the Volga Bulghars by way of Central Asia, in response to the latter’s request for funds and instruction in Islamic ways.

Though the desired alliance between these people and the Abbasids failed to arise, his descriptions of the journey, and the life and customs of the people he encountered (including ones of likely Scandinavian origin), provide an invaluable and rare account of the conditions of a disparate group of people.

#Ludovico de Varthema#

This Italian traveler left Europe in the early 16th century and journeyed across much of the Middle East, in particular being one of the first non-Muslim Europeans to enter their holy cities of Makkah and Madinah. He proceeded then towards western India, visiting Gujarat, Bijapur and the Malabar coast, which is described in much detail. Ceylon, Bengal, and Southeast Asia are next on his itinerary, after which he returns by sea to Europe and in due course publishes his account of his travels entitled Itinerario de Ludouico de Varthema Bolognese.

 
There are certainly other such accounts though. These are only the ones I'm familiar with.