The Thor Amulet is at the Great Wall of China (1593 AD) at the upper right, where you must play a memory game to get it. It goes to the Vikings at 831 AD.
The silver Peace Medal is under a wooden bridge at the workshop of Leonardo da vinci (1516 AD). Climb the platform that is supported by ropes and move to the one on the left. This raises a small platform down in the "hole" in the walkway. You can jump down and across to the medal. This is slightly easier if you have the Glider Wings that Leonardo gives you for returning his notebook. The Peace Medal goes to Lewis and Clark at 1805 AD.
you leave and go to a different island
After you travel back to the future to get the Island Medallion, return to the present to leave the island (just as you did after completing the items), by using the Lab selection on the time device.
Use the Lab setting on the time device, just as you did to get to the restored future.
Once you have your Island Medallion, simply use the time device again to go to Pendulum's Lab (12 o'clock).
You leave the island the same way you returned to the future the second time, by using the Lab setting to return to the "present time", outside Pendulum's Lab.
Once you have the island medallion, you can return to the present using the time device on the "Lab" setting (12 o'clock), as you can at any point during the game.
On your first visit using the future machine, you get the time device from your future self, so that you can restore the items that are tangled in time. When you finish and get the Island Medallion, you still have the device. Set it to "Lab" and go back to Professor Pendulum's lab.
After you get the Island Medallion, or anytime you want to leave the island, use the time device at the "LAB" setting, which takes you outside Pendulum's Lab in the "present". Then you can leave Main Street as usual in the blimp.
You have to go back through the Future machine to reach your future self in the repaired timeline. Make your way to "your" sky home in 2059. This is where you get your medallion, from your "future self". (see the related solution question)
When you return an item on Time Tangled Island, the time device shows "green" for that time (there may still be objects to come from there). When all times are green, travel to the Lab (12 o'clock. the present). Use the Future Machine to travel to the restored future and get you Island Medallion.
The oracle is in the building on the hill on Time Tangled Island.
Once you get into your sky home in the future, go to the top right corner, where you will meet yourself (you in 50 years). You will then receive the Island Medallion. (see related question)