The replacement for synthetic underwear is wool, not cotton. For weeklong wilderness backpacking trips, I used to vacillate between a fresh pair of synthetic boxer briefs every every day, which took a lot of space, and one for every other day, which was gross. Now I wear a single pair of wool boxer briefs all week, and it feels and smells better at the end of the week than synthetic underwear does after two days. Same for sleeping: I used to bring a couple of pairs of synthetic boxers, and now I'm down to one pair of wool underwear. Two pairs for the entire week, one for days and one for nights.
Granted, wilderness backpacking has completely different standards for smells and grossness, but the comparison carries over to the higher standards of the "front country." In hot, sweaty weather, synthetic underwear gets noticeably grosser and smellier than wool over the course of a day.
Mine are SmartWool brand. According to their web site, they use an 88% wool, 12% nylon blend, but I haven't experienced any odor issues.
I have an older pair that is 100% wool (I don't recall the brand; they might be a very very old SmartWool product) but the lack of stretch makes them less comfortable.
Editing here since my original comment is too old: in a pinch, I've also had much better results washing wool underwear in a hotel bathroom than synthetic underwear. Contrary to their reputation, neither wool nor synthetic underwear dry quickly, even with a hair dryer, but wool dries faster, feels cleaner after washing, and in the worst case scenario is much more comfortable wet than synthetic underwear.
Granted, wilderness backpacking has completely different standards for smells and grossness, but the comparison carries over to the higher standards of the "front country." In hot, sweaty weather, synthetic underwear gets noticeably grosser and smellier than wool over the course of a day.