To reach Odym, you must first complete Green Loontern's quest in The Watchtower to unlock the Moon Base. From here, you can build up several Lantern gates to reach the various planets.
Gold Brick #1
Talk to the Green Loontern by the teleporter to get a tour of the planet. Follow him around, taking out the beetles that attack. When the tour ends, you'll receive a brick.
Gold Brick #2
You can also find Aquaman on this planet; he needs help returning a crocodile to a safer place. Follow him and the croc, and punch the green guy in the head any time he stops to snap at civilians. When the croc returns home, you'll get a brick.
Gold Brick #3
You'll find Gorilla Grodd on Odym too. He needs help finding a banana for his ultimate weapon. It's a little tough to find, but you'll have to scour the area to find this tree and sensor spot. Use a sensory character to reveal a dirt patch, use a digging character to pull up a handle, build it onto the tree, and finally grapple it off the tree to shake a banana loose. Return this to Grodd to complete the quest.
Gold Brick #4
Grodd has another quest for you: dealing with his ex, Giganta. To defeat her, simply hop into the banana cannon and shoot her until she's defeated. You'll earn a brick for it.
Gold Brick #5
For the final quest, you'll have to make amends with Giganta. To do this, follow her and take out 30 bugs as they appear. Once they're all defeated, you'll receive a Gold Brick.
Gold Brick #6
The rest of the Gold Bricks can be found underneath the beacons in the sky, one of which points to a flying challenge. To win, simply fly through all the rings before time runs out.
Gold Brick #7
There's another flying challenge above a rock arch. Zip through the rings just like the first time to win another brick.
Gold Brick #8
There's a footrace to complete here too. Dash through all the gates before time expires for a Gold Brick.
Gold Brick #9
One beacon points to the crashed Brainiac ship. Inside and around it are six of these blue engines; destroy them all to kill the lasers surrounding the Gold Brick inside the ship.
Gold Brick #10
The village area (also marked by a beacon) holds 14 - yes, 14 - flags. Bust one up, then scan the rooftops to find the others and bust them before time runs out. You'll win a Gold Brick if you're successful.
Gold Brick #11
One beacon points to this flower, which you can bust up to reveal dirt patch. Use a digging character to pull out another brick.
Gold Brick #12
Finally, a beacon will point you toward this group of silver rocks. Use a demolition character to blow away the top one, revealing a switch. Step on it to open the chest below, which holds a tinier silver box. Blow this up too to find a Gold Brick inside.