I held back for the longest time from sugar-free drinks since they tasted worse to me and I didn't want to give up sugary sodas. Then, last year, I tried to go exclusively sugar-free for just a few weeks. Then the weeks turned into months, and now 1 1/2 years. The surprising thing to me is that I don't miss the sugary drinks at all as the sugar-free alternatives are so good these days that they cover all my cravings.
This thread, though, is a good example of why people are skeptical to sugar-free alternatives. Bogus science, "I heard somewhere", and fear of the unknown all contribute to a feeling that "if so many people say it's bad, it's got to be bad, right?"
The truth is, few foods have been as meticulously researched as sweeteners, and they have all been cleared as safe within surprisingly large margins. According to most studies I've seen, you should be able to consume 4-5 liters of drinks containing aspartame every single day for your entire life without any negative effects. Honestly, you have way more reason to be concerned for your dental health at that point.
Some people have minor reactions to certain sweeteners, though, but that goes for just about anything that is edible. If you find yourself getting headaches or loose bowels, try a different sweetener.
The one sweetener you should be careful with is sorbitol, however. It is a laxative to almost everyone, and some people get that effect the moment they get anything with sorbitol inside their bodies.