I cannot recall how many cotton shirts I have worn out over the years wearing them backpacking in the summer. In fact, I believe the whole cotton kills mantra only recently started in the early 90's or so.
I have found that cotton-poly blend shirts work great for summer - the best of both worlds - more comfort, less smell than polypro and still dries a bit quicker than cotton. The cotton/poly t-shirts are pretty easy to come by as well.
For shorts, I usually wear just a cheap pair of nylon Columbia brand workout shorts (I think they are called river shorts).
Many old timers wear blue work pants that are cotton poly blend (like a garage mechanic would wear). They are cheap, dry quickly and are pretty comfortable as well.
I am re-editing this, I should be more clear - I've worn cotton for quite a while now in June, July, August or September - the periods when it can be sweltering hot. Of course, I keep my eye on the weather for the day - -If it is going to be changeable or wet, or I am in the High Peaks, I'll wear my polypro and whatnot. For backpacking trips, I now longer wear cotton, as it is just too heavy to carry around, and as mentioned, it is hard to dry.
Sorry, I should have been more clear.