new cassette....new chain rings?
When replacing cassette (and chain) does it follow that chainrings should be replaced as well?
It depends on how well you've looked after your drivetrain. You only need to replace the cassette if the teeth on chainrings are noticeably worn down. If the teeth are in good shape, then use the old one. See the link for examples.
Why are you replacing the cassette and chain. If it's because you let your chain become elongated, then you will need to replace at least your "favorite" ring. If you're a "middle ring only guy" then just that ring will probably need to be replaced.