This multi-center study will evaluate whether thalidomide can improve the effectiveness of the drugs leuprolide or goserelin in treating testosterone-dependent prostate cancer. Leuprolide and goserelin-both approved to treat prostate cancer-reduce testosterone production, which, in most patients, reduces the size of the tumor. Thalidomide, a drug used for many years to treat leprosy, blocks the growth of blood vessels that may be important to disease progression.