Tusheti is a region in north-east of Georgia, formally a part of Kakheti. It is famous for its beautiful mountains accessible to hikers and has an eponymous national park.
The hiking path from Omalo to Shatili is mostly accessible from one of the ends.
The road from Tbilisi to Telavi is fairly OK to hitchike, as well as the small road from Telavi towards Alaverdi Monastery. In summer, crowds of desperate hitchhikers stand on roads around Alaverdi and Pshaveli, often waiting hours without result. The dirt road from Pshaveli to Omalo is very long and difficult and mostly used by commercial jeeps. Either be very patient or prepare a substantial amount of money or one or two days for the hiking tour from Pshaveli to Omalo.
A tar road from Zhinvali to Shatili is much easier for hitchhiking, but still very scarcely used. In Shatili, it is a good idea to wait at the (only) exit from the village, on the other bank of the river (opposite to the old village complex). There are minibuses departing quite regularly, as well as some tourists in smaller cars. Still, some people report waiting a whole day without hitching a ride, though.