Hi,
The only market I know of is a fruit,veg & fish market in Rhodes town on a Wednesday and Saturday morning. It is not in the town centre itself but just along from the cemetary on the Kallithea to Rhodes road. Starts 7.30am finishes 2pm. Beautiful home grown produce at excellent prices.

I agree with the previous post the only regular markets you'll find on Rhodes are for food.
If you're looking for souvenirs then try Rhodes Old Town or the 'New Market' which isn't really a market as you'd picture it but you will find sovenir shops there.

Oh really. How dissapointing. I was really hoping to bring back some local souvenirs. Oh well we are going to the Marmaris market so will have to do that there.

Thanks for the posts x