How do I know when my chainrings are worn out?

Chainrings can show signs of wear even pretty early in the life cycle.  A few bad shifts, or a little mishap can cause damage to the black anodize and let the shiny aluminum show through.  These are typical of used rings, but it does not mean the ring is worn out.
Worn Out Chainring
On the other hand, many rings are run far beyond when they are “worn out”.  Seriously warn rings will show deep cupping in the areas between the teeth.  On WickWerks rings, the anodize will disappear in the troughs between the teeth.  These are indications, but the real test is function.

Often a chain that is long worn out will have grooved itself into the chainring in such a way that it all still works (mostly), but if the chain is replaced, the new one will not function well.  This is a bad situation because it indicates replacement level wear on the chainrings, but also probably means the rear cassette needs to be replaced.

Summation:  If the chain measures in spec, then until you feel a significant degradation in shift quality or some other aspect (such as chain skip), then keep riding them.