Looking on thesaurus.com I can find only synonyms for "desert" with negative connotations. Are there any synonyms with positive connotations? Specifically, something that invokes the sense of clean desolation and unspoiled nature.
The Etymology tells the story:
The Classical Latin word deserta (abandoned, deserted wife) is derived from the Latin word desertus (deserted, uninhabited, without people), which is derived from the Latin verb deserere (to cease to be concerned with; to fail, fall short; to leave, depart, quit), which is derived from negative or past de- plus the verb serere (to plant, sow).
Hence, a desert is unused or unusable land. Arabia Deserta is the classical Latin name for guess which country? And guess what it means?
There really aren't many positive connotations available for the word desert; perhaps that's one reason why we often spell it dessert and vice versa. There's rarely any confusion about which word is meant.
Following your link shows a tremendous number of synonyms with negative connotations! Here are some relatively neutral, albeit not quite synonomous, words:
Expanse, as in wide expanse or open expanse, is fairly positive. It evokes a sense of opportunity and possibility, to me: Riding off into the open expanse.
expanse A wide and open extent, as of surface, land, or sky.
Rather than look for a word with positive connotations (even National Geographic wasn't able to put their usual positive gloss on an article about the Atacama), you can use a word that gives an exotic flavor to the defining feature of a desert:
xeric, adj.: (of an environment or habitat) containing little moisture; very dry [OED]
So you can write about crossing the xeric landscape of the Mojave, for example.
The Thebaid or Thebais (Greek: Θηβαΐς, Thēbaïs) is the region of ancient Egypt containing the thirteen southernmost nomes of Upper Egypt, from Abydos to Aswan. It acquired its name from its proximity to the ancient Egyptian capital of Thebes. During the Ancient Egyptian dynasties this region was dominated by Thebes and its priesthood at the temple of Amun at Karnak. [...]
Around the 5th century, since it was a desert, the Thebaid became a place of retreat of a number of Christian hermits, and was the birthplace of Pachomius. In Christian art, the Thebaid was represented as a place with numerous monks.
The subjects range from the pleasures of courtly life to the spiritual rewards of religious retirement from the world in a Thebaid, to the degradations and miseries of the poor [...]