I have three recessed lights in my kitchen, on a 3 way switch they have worked fine for over 20 years. The other day they would not turn on. I replaced the switch but nothing. I'm getting power to the switch.

Having one 3 way switch means there is another, any other switches that also control this light would be a four way, so there should be other switches to check.

If you replaced the 3 way switch, you need to be careful to put the wires back exactly, the one dark terminal labeled common is the important one.

How many screws and what color wires went to the switch you replaced? Tell us the same info for the second switch. How did you decide which switch to replace?

There are 2 switches on a 3way. Did you replace both? Sometimes a circuit won't work because 1 of the switches is not fully off. Probably is a switch problem since I doubt the 3 lights are on a single breaker. Could be a squirrel or other animal problem gnawling the wire. Could be a loose hot wire or neutral in the circuit. Check all the wiring is tight on the switches and wirenuts are all tight too.