Konye-Urgench is a city in Turkmenistan.
East towards Dashoguz
You can hitch-hike past the shared-taxis stand 2kms East from the mausoleum complex.
South towards Ashgabat
The junction with the main road is about 90 kms South of Konye-Urgench, at the small town of Ruruhbellet. The road to Ruruhbellet is in very bad condition and traffic is scarce.
You can take a minibus (2.60M, less than 1$, in april 2012) to Ruruhbellet and hitch-hike from there. The minibus leaves 2 or 3 times during the day from a bus stand in the Southern part of town.
If you want to hitch-hike all the way, consider backtracking to Dashoguz until the junction with the main road to Ashgabat, or try your luck after the Turabeg Khanum Mausoleum, 3kms South of the city center.