Thanks for all the tips. It's a new (= month old) cog and chain, but old chainring. I've checked the chain and it hasn't worn, so I don't think the chainring is too worn. I've tried changing the chain to a brand new one, and that doesn't make the chainring click, so I don't really know what the problem is. I tried loosening up the chain tension, didn't make a difference.

I guess I'll just live with it, since there isn't anything obviously wrong other than the noise itself