Step 1: Understanding the Virashaiva tradition.
The Virashaiva or Lingayat tradition emerged in Karnataka during the 12th century. It emphasized devotion to Lord Shiva and rejected caste distinctions and ritual practices.
Step 2: Nature of Vachanas.
The followers of this tradition, like Basavanna, Akka Mahadevi, and Allama Prabhu, composed short poetic sayings called Vachanas. These Vachanas were written in the Kannada language, making them accessible to common people.
Step 3: Conclusion.
Thus, the Vachanas of the Virashaiva tradition were composed in Kannada.
\[
\boxed{\text{Kannada}}
\]