The first thing that we need to define is what constitutes "emitting fumes."
Generally, when it comes to indoor air quality, what we worry about are things called volatile organic compounds, or VOCs. These are naturally occurring compounds which originate in plants; however, the ones that affect our health and indoor air quality are the man-made variety. A major source of man-made VOCs are solvents, especially paints and protective coatings, so when it comes to any flooring, wall surface, or anything else that you are putting in your home you need to be diligent in assuring that the coatings (paint, sealer, primer, etc.) are aqueous in nature and do not contain significant amounts of VOCs.
The most common VOC that we find in homes, which ironically enough has one of the higher potential long-term negative impacts on human health, is urea formaldehyde. Urea formaldehyde can be found in many products such as adhesives, finishes and manufactured or molded products. You will find significant levels in the backing and core of many manufactured flooring products, including some cork and bamboo (also in cabinetry or furniture that is not solid wood; i.e., particle board or MDF).
So what this means to you is that you need to be diligent in doing your homework on where your cork or bamboo comes from and what it is made of. Avoid the "cheap" lines of either product, since they will fundamentally be laden with formaldehyde. If indeed you are adverse to hard, cold surfaces, cork will certainly be your best choice. Resist the urge to shop on the internet for the cheapest material. It will likely come from China and be filled with offgassing chemicals.
I recommend a product called EcoCork from a brand called Natural Cork, which is a division of US Floors. This product is made in the United States and the manufacturer can tell you everything that is in the product and supply you with an MSD sheet (material safety data) which will list all the ingredients in the product. They have reasonably priced products with fantastic, long-lasting finishes which will have nominal offgassing; nothing to worry about compared to MDF or particle board products, which contain hundreds of times more potentially unhealthy chemicals.
Also, make sure you use a reputable installer who knows your concerns regarding indoor air quality. All it takes is one tube of caulk or adhesive that contains high VOCs to completely defeat all your research and the money that you spent on your nice natural-cork floors.
For more information:
Check thecork flooring pros and conssection of GreenHomeGuide's green flooring backgrounder article.
In our Q&A section, read architect Ian MacLeod's "I'm concerned about toxic offgassing from OSB subflooring and roofing. Are safer products available?"
AndPeter Kellner's Q&A "Can you offer tips for comparing the indoor air quality effects of different cork flooring products?"