Mars Prize #1
Map: Hall of Heroes
Prize: Mystic Urn

This button is located near the beginning of the level in the red carpeted hallway with the power crystal and a Medusa.

Mars Prize #2
Map: Hall of Heroes
Prize: Krater of Might

This button is hidden behind an illusionary hedge in the hedge maze. Simply walk through the hedge to find it. It's near the top of the maze (as you enter from the gold teleporter).

Mars Prize #3
Map: Forum of Zeus
Prize: Tome of Power

This one is kind of tricky to get to. It's on the ledge running around the courtyard containing a fountain, a Medusa, and part of Weapon #4. Creep out slowly (turn "always run" off) from the window overlooking the courtyard to get onto the ledge.

Mars Prize #4
Map: Forum of Zeus
Prizes: Ring of Turning, Staff of the Ovinomancer

When you hit the button that raises stairs from the ground and drops a platform outside by the pool, it also opens the way to this button. Turn the corner and walk up to the darker gray wall to swing it open. Kill the spider and find the button.

Mars Prize #5
Map: Gardens of Athena
Prizes: 4 Glyphs of the Ancient, Boots of Speed

This is a two-stepper. First, you have to dive into the pool near the breakable stairs (shown in the screenshot). Break the wall and hit the button. this will drop a platform down into a second pool in a nearby courtyard, granting you access to the Glpyhs and the Mars Prize button.

Mars Prize #6
Map: Gardens of Athena
Prizes: Force Cube, Summoning Stone

This is hidden in the Temple of Athena behind the statue where you find the gold bar. The top of the steps leading into the temple is cracked and hides a floorplate. To avoid awakening the Imps, stand outside the room and fire at the cracked step with a ranged weapon. Step on the revealed floorplate, which opens a secret chamber behind the statue.

Mars Prize #7
Map: Baths of Demetrius
Prizes: Ring of Regeneration

This is just outside of the room where you have to press the fire, water, steam, and air buttons. Break the cracked wall and follow the crawl-space to the ring and button.

Mars Prize #8
Map: Baths of Demetrius
Prize: Chaos Device

This is near the end of the Baths of Demetrius level, just before you get the Crown. Swim down to the left of the platform shown in the screenshot. Follow the underwater tunnel to a steam room containing the button.