https://gamebanana.com/sounds/61710 - mod created by PimpUigi
This mod reimplements the removed Michael Jackson team soundtracks (and others) from Sonic 3. These are:
Original Invincibility Theme - regular Sonic 3
Knuckles' Theme - MJ's Team
Carnival Night Zone - MJ's Team
Ice Cap Zone - MJ's Team
Launch Base Zone - MJ's Team
Launch Base Act 2 Complete - regular Sonic 3
Doomsday Zone Complete - regular Sonic & Knuckles
along with a remix of the new Super Sonic theme in Sonic Origins to make it sound less obnoxious

There are two downloads - the first being "thedefinitivesonic3ost.zip".
There is also a second download present which adds the Sonic 3 credits music before the staff roll of Sonic 3 and Knuckles (the cutscene where the master emerald is recovered.) If you want this version you will have to download "thedefinitivesonic3ost_alt.zip".

The HedgeModManager is a modding tool for Sonic Origins and other games like Sonic Generations. If you'd like to install more than one mod for Origins, I recommend that you download it.
Go to https://github.com/thesupersonic16/HedgeModManager and click on "releases" to the right. Then click on the HedgeModManager.exe to download. The latest version, 7.8, adds Origins support.
Once you've downloaded it, open it and it should work. Click "Add Mod" and then find the Definitive Sonic 3 OST.zip you downloaded. Then tick the checkbox next to the mod. The interface should look like below:



Then click on save and play to open the game!

If you don't want to install HedgeModManager, you can just drag the music files direct.

If you don't have a zip file manager, you should get either WinRar or 7Zip. I use 7Zip.
Open the zip file and navigate through so the two music files, "Music03_3SK.acb" and "Music03_S3K.awb" are visible.
Then navigate to your steamapps/common directory. Find the SonicOrigins folder.
Navigate to image, x64, raw, and then sound. The sound folder should look like this.

Replace the music files in the folder with the ones you have in the zip.

The easiest way to test if the mod works is by going to the Mission menu, and selecting the S3&K mission "Knuckles' Climbing Challenge".

You're done! Now enjoy that MJ goodness.