Trekking in Tibet is in its infancy and it is likely to remain that way as long as regulations are so cumbersome and unpredictable in Tibet. However, there are two routes that are fairly well established, the kora (cirumabulation) around the sacred Mt. Kailash (the holiest mountain for Tibetan Buddhist and Hindus alike) and the trek near Lhasa between Ganden and Samye monasteries. Unfortunately regulations have gotten considerably more strict since I was trekking in Tibet back in 2007, and it is now virtually impossible to travel in Tibet independently much less trek. A loosing of these restrictions seems unlikely in the near future.
Best Season: May-September
Base Cities: Lhasa (Trains and Flights from Beijing, Jeeps from Kathmandu)
Guide book/Maps: Lonely Planet Tibet
Alternative Trekking Locations
If you do want to trek in culturally Tibetan Areas Independently there are a number of alternates to the strict regulations of the Tibet Autonomous region
There are several areas in China that are essentially historically Tibetan but fall into other Chinese Provinces. Areas of Sichuan, Gansu, Qinghai, and Yunnan all have Tibetan areas. The area around Ganzi in Sichuan is particularly scenic and would have good potential for trekking though I've only done day hikes in the area. The northeastern corner of Yunnan, near Deqen is another area though I have not been there personally.
Ladakh is the one of the easiest places with Tibetan culture to trek independently and has some spectacular scenery as well.
Spit Valley just south of Ladakh in Himachal Pradesh is another Tibetan region within India which has number of monasteries and impressive mountain scenery.
Sikkim and Arunachal Pradesh are options in Northeast India though these areas also have regulations, though not as burdensome as Tibet.
The areas in Nepal most similar to Tibet, Dolpo, Upper Musatang, and Nar Phu, all required restricted area permits which basically puts them off limits to independent trekking. However all of the treks in to the high Himalayas of Nepal will give you a taste of Tibetan culture, as most areas are heavily influenced by Tibetan religion and culture.