I am hypothyroid and likely had been for a number of years before I was diagnosed about 5 years ago. It depends upon why you're hypothyroid as to whether supplements may help, but it's really not something you should do on your own.
Thyroid hormones are made up of Tyrosine and Iodine, so a natural approach would be to add those to your diet. Also, selenium and B vitamins are important for thyroid function.
Also, the thyroid prescription you might receive isn't really a drug. It's a hormone (synthetic or natural) that is replacing the hormone that your body may not be making (if you're hypothyroid, which is the most typical). The synthetic hormones are Synthroid and Levoxyl, both of which are just T4. Cytomel is a T3 synthetic. The most popular natural prescription is Armour Thyroid and it's better in that it also contains T2 and T1 and calcitonin. The latter is really important for bone building. Many people do not have complete resolution of hypothyroid symptoms unless they take something like Armour. However, some folks have allergies to Armour since it is derived from pork.
There is a Texas Thyroid Yahoo group that you should join and post your lab results (always get a copy) to get feedback from folks that have been dealing with this for a number of years. There's at least 100 years (if not several hundreds) of thyroid hormone replacement therapy experience on that list: http://health.groups.yahoo.com/group/Texas_Thyroid_Groups/
Your doctor should have measured your antibodies and if you have antibodies, your thyroid problem is autoimmune which means you should be screened for celiac disease (many doctors miss this and this is actually one of the most important things for them to screen as it's believed to be the root of most auto-immune diseases).