Hi Cassie88,
Congratulations on your new mattress 
It would be nice if you could mention who you bought from so that others from your area can benefit from your experience and feedback as well.
As you have discovered … “bamboo” sheets are usually made in either a blend with cotton or polyester. They are very soft and wick moisture very well and are strong and breathable. I should also mention that bamboo is a form of viscose fiber (like rayon) and there is somewhat of a greenwashing story attached to them. Cellulose/viscose types of fibers like bamboo, eucalyptus, wood pulp or others are neither natural or synthetic and are more “in between” or artificial. A google search on “greenwashing bamboo fiber” will bring up many links including here and here and here (and the further links they lead to). Post #11 here also has some good information comparing cotton to bamboo. There is also a good article here about the many types of viscose fibers (such as bamboo, beech, eucalyptus, and other regenerated cellulosic fibers).
This webpage about different types of cotton sheets and how they compare to others (including bamboo and other viscose fibers) also includes some great information and some specific recommendations and links to other good sources of information as well and would be well worth reading for those that are looking for good quality/value cotton sheets.
They also have a very good article about linen sheets here as well.
Post #26 here along with post #3 here also has more information and links to more resources and information about sheets and bedding.
The “true” thread count of a fiber and the weave pattern it uses are also an important part of the breathability, strength, and feel of the fabric. There is some good information about thread count here and here and about thread counts, weave patterns, and fabric types here and also good explanations of the various types of fibers here and some of the pros and cons of each here.
While I am no expert on bedding … if I was going for the “best” combination of coolness, durability, feel, natural sources and methods, and “value” … I would be leaning towards linen (flax). Silk would be one of the most cooling fabrics as well but both of these are in higher budget ranges.
In the same way as mattresses … I would also tend towards purchasing from outlets and manufacturers that are the most knowledgeable and open about the different fibers and the pros and cons of each. There is just as much misinformation “out there” with various fabric materials as there is in mattresses. For me … if certain qualities were important to me and I really wanted to know that what I was buying would perform in the way I wanted it to … I would make who I was buying from and their level of knowledge and the information and education they provide just as important as the actual product choice I was making. Making choices based on the “misinformation” or misleading specs that are so common may lead to a “price” that initially seems to be better but it can also lead to buying something that doesn’t have the qualities or “real specs” or materials you are looking for and what you believe you are buying may be different in either actual materials or performance than what you were hoping for. In some cases what you buy may be OK but because you didn’t know exactly what you bought you may also not ever know what you are missing and “could have bought” (that could have “worked” for what you wanted even better).
A few examples of outlets like this (among many others) that have a variety of products and knowledge about different fibers and fabrics and the differences between them are here and here and here. These are of course just some examples of many others who are more about helping their customers make better choices and are focused more on real value than on price alone.
Of course I haven’t done any extensive price or “value” comparisons but having good information so you know you are really comparing apples to apples (instead of “real” apples to something that “looks like apples”) can make “value shopping” much easier and more effective.
Hope this helps.
Phoenix
PS: I will add some of the other posts and threads in the forum here from time to time so there is one central post that links to many of the better or more interesting posts and topics about sheets.
There is some interesting discussions about silk and linen sheets in this thread and this thread. There is also a review of the Linoto Belgian Eco-Linen sheets here.
Some feedback about sheets that some of the other members are sleeping on and like is in this thread (and I encourage any of the members here to add their own feedback about sheets they are sleeping on and like).
There is some information and options about sheets for thinner mattresses in post #3 here
Some discussion about cotton percale sheets is in this topic
Some information and feedback about Intellatex sheets is here.