We are going through the same thing - my daughter turned 3 a couple of weeks ago and on her 3rd birthday just decided to stop napping, after having been a pretty consistent napper her whole little life. I fought with her for two days, taking two hours to get her down for a nap that then went until 4.30 or 5 and caused her to stay up until 11, and then I gave up. We do have quiet time, sometimes together reading stories, sometimes on her own, or sometimes I let her watch a video. (I A. 8 months pregnant, so I still need MY nap!) I've learned not to go anywhere in the car late in the day, because then she immediately falls asleep - sometimes, if she is really a pill and it's before 3 I will take her for a short ride in the car and then let her have a half hour car nap. She's really grumpy between 4 and 6 pm and then she perks up again until bedtime (which we moved about 1 -2 hours earlier, 7.30 or 8 instead of 9.30 or 10). I try to keep her outside as much as possible in the afternoons and evenings while the weather is still good. ALl of that said, we're falling into a routine now with the earlier bedtime, and though she still doesn't want to go to bed even though she's practically falling over with exhaustion, once she falls asleep she's sound asleep for a good 11 hours. I think it's up to you which path works better for your family.
Oh, one other thing that helps - PROTEIN! The better the quality of the food she eats, the better her mood is in general. So I've been trying to feed her more eggs and turkey snacks and peanut butter and things like that, then the afternoons don't seem as hard.