Poseidon's Ring
Map: Hall of Heroes
Prize: Ring of Waterbreathing

Look behind the statue that greets you shortly after you start this hub. There's a ring tucked back there.

The Brown Hedge
Map: Hall of Heroes
Prize: Force Cube

This is in the hedge maze near the corner containing a button and a medusa. Destroy the dead hedge.

The House Roof
Map: Gardens of Athena
Prizes: Krater of Might, Ring of Flying

Pretty easy. Just grab the ring hiding over in the corner of the courtyard near Mars Prize #5. Fly up and grab the Krater.

The Temple of Athena
Map: Gardens of Athena
Prize: Stone of Summoning

While inside, step over to the right to read the message. This will illuminate the button on the ceiling. Look up and shoot it to make the Stone appear on the alter.

Storeroom #1
Map: Hall of Heroes
Prize: Chaos Device

The storeroom is located underneath the building where you placed the Power Crystal. It's opposite the archway leading to the Baths of Demetrius. Break the cracked wall shown in the screenshot.

Storeroom #2
Map: Hall of Heroes
Prize: Staff of the Ovinomancer

In the same area as the secret above, smash the cracks up on the wall. To get up there, you'll need to push a barrel over, so be careful not to break it! Be prepared for a fight, too. This secret lies in the middle of a giant spiders' den!