Hi Nemix,
Both Tencel and Bamboo are cellulosic rayon fibers that are made from regenerated cellulose that comes from different sources. They can be made using either the original viscose process or the newer lyocell process. Tencel is Lenzing’s trade name for a fiber that uses eucalyptus trees as its source of cellulose and the lyocell process to make the fiber. Cellulose that comes from bamboo can be made with either the original viscose process or the lyocell process and uses bamboo as its source of cellulose. There are also many other variations that use different sources for the cellulose (such as beech trees) that are made into other versions of rayon as well (such as modal).
The cellulosic raw materials in the lyocell process aren’t any more natural than the viscose process (they are both semi synthetic fibers) but it is more environmentally friendly and improves the wet strength, the feel and softness/smoothness, and reduces shrinkage of the fiber.
There is some good information about different types of rayon fibers and production methods here and here and here
There is also more information about sheets and bedding in post #7 here and the other posts and sources of information it links to that may be helpful as well.
Phoenix