The four Items of Power are listed below, and must be collected in the same playthrough for the achievement to pop.
Portal Horn - Island 1. You'll need 4 gems and 5 coins to complete this puzzle, so you'll need to travel to at least one other island before unlocking this ability. First, find an area with separate sun and moon symbols engraved on a couple of large stones. Here, you'll need to spend 1 gem and 4 coins to acquire the Day/Night horse. Then, you'll need to find an area with two tall statues that are crying. While on the Day/Night horse, spend one gem in the morning on the left pedestal, one gem during midday on the middle pedestal, and one gem in the afternoon on the right pedestal. Once you've spent your gems, the statues stop crying, and the Portal Horn descends from above to claim for one coin.
Mjölnir - Island 3. You'll need about 10-15 coins to complete this puzzle. I'd recommend bringing about 25 coins or so for insurance so you don't have to go back and forth from town if the statue is particularly far away. Find a statue of a man with a viking helmet holding a hammer (that's Thor) standing to the left of seven switchable statues with runes engraved on them. You'll be looking to make the following combination, in order, using those seven switchable statues: ᛣ ᛁ ᛅ ᛂ ᚾ ᛁ ᛦ
If the above runic text doesn't display, then find the two huge boulders with three runes etched on each. You'll need to copy those runes in sequence on the switchable statues. Once you've got the sequence punched in, stand in front of Thor and spend 7 coins. If the sequence is incorrect, the runes on the switchable statues will glow red briefly. Adjust accordingly, and once you've got the right sequence, Mjölnir descends from above and can be claimed for one coin.
Side Note: In my experience, the runic code doesn't change between playthroughs. I can update if someone discovers a different code, though.
Trophy Head - Island 4. You'll need 4 archers with shields, 1 squire, and 14 coins to complete this puzzle, so make sure you can spare the resources. There are 4 flame sconces and the bust of a woman's head; the right half of her face is skeletal. You'll need to light each of the sconces for 2 coins apiece and pay tribute at the woman's head for 5 coins. The archers will drop off their weapons at each of the sconces, and the squire will drop off his weapons at the woman's head. After a brief animation, the Trophy Head will descend for you to purchase for one coin.
Loki's Staff - Island 6. You'll need nine coins using this strategy to complete this puzzle. I would recommend doing this in coop mode, so if the worse comes to pass, your game won't immediately end – the second player can stay in the village while you try this out. Also, make a save before trying to unlock this. I would recommend emptying your pockets before you try this, but to each their own.
You're looking for a statue of a man flanked by two serpents. You're going to have to lose your crown in order to unlock this ability; hence, coop mode and the save. The idea is that when you are attacked by the Greed, and your crown flies off your head, the crown's trajectory will intersect near the statue's hands. If done correctly, the crown will go to the statue's hands, a brief animation will light up the statue and a staff will appear for you to claim for one coin. You will also be turned into a Greedling. Fun!
Go back to town and have the second player pay 8 coins to restore your crown to your head (after you've reverted back to being a human; your townsfolk will attack you on sight, which breaks the spell). Head back to the statue with a coin in tow, and you can get yourself Loki's Staff, and complete this achievement.