My 3 year old son still naps, everyday.
We have the SAME routine about naps, that I have had since he was a baby. The only thing that changed, was his nap time and frequency.
With him only being on 1 nap a day, now at his current age.
If he does not nap, he can't last the whole day, and is tired, fussy, or over-tired, which being over-tired can also make a kid "hyper."
I tell my son, "nap soon... " then he knows and gets in his routine. He likes to watch a "Little Bear" show first, I give him a cup of milk. Brush teeth, then to bed. Same as the routine for night-time.
He knows it like auto-pilot. He will even tell me sometimes "I'm tired... I'm going to bed..." and then he goes into the routine.
I "read" while in the room, and he lays down... I put on a fan for white-noise. And then he winds-down.
Then falls asleep.
The "rule" is he has to lie down.
Or he likes to, and I allow it, that he likes to play his Leapster first, before sleeping. It winds him down. But I only allow it for 5 minutes. Then he will put it away, himself. Then he settles himself and sleeps.
He naps for 2 hours. Sometimes more. And at night, he sleeps fine.
His naps are either in the morning... or after my daughter gets home from school in the afternoon.
Yes, once in awhile he does not nap. But that is rare.
Yes, 3 years old is harder than 2 years old. Many things changing in them developmentally.
Skipping naps... actually yes, makes them over-tired... and when over-tired kids get unpleasant. Or, they cannot sleep well at night.. because they are over-tired.
My daughter, who is now 7... will still nap occasionally. BOTH my kids just woke up from a nap, actually.
I just do the SAME routine for "naps"... everyday... the same routine as for bedtime.
It takes consistency.
But its not easy... but I've been doing the SAME nap/sleep routine with my kids since my daughter was born... then when my son was born... the SAME every time. So for them, it is like a "normal" every day thing. Not just willy-nilly.
I don't force it though. But, I maintain the routine... and they have been cooperative.
All the best,
Susan