What can I use to remove these stains?

[Truckingonandon]

Non removable chair seat cushion covers. I've tried everything but to no avail. Anyone got any magic tricks? It's basically water stains and dust, rather than anything untoward.

These type of chairs are very easy to recover. I would think along those lines. Sorry.

Dr Beckmann do a carpet and upholstery cleaner which cleaned our dining room chairs well. They are grey though which is a lot more forgiving than your paler ones. Worry a shit though as it's not expensive.

Worry a shit?!
Worth a shot...

If you do want to recover them you can buy the fabric, cut a template a couple of inches bigger than the seat all round, and with an industrial stapler (around 10 quid) staple the new fabric underneath to the chair stretching it as you go. Have a go with a cheap bit of old sheet to get the technique. It is very easy.