Everyone I know does this, including the strictest. That is, if ordering a cold drink at a non-kosher restaurant or bar, they will have it in the glass the institution provides, not ask for it in a disposable cup. Why is this, if Ashkenazim are supposed to be machmir regarding absorption of taste in glass?
(This page says that drinking from the glass is permitted only derech arai--on an occasional basis--and because the glass is presumed clean and to have only been used with cold drinks. If these presumptions are enough that the glass can be considered kosher, then I don't understand why derech arai would apply; even less if these presumptions are not enough to consider it kosher.)