Green Day: Rock Band

You can also get this on Easy (or any difficulty) with 2 people and 2 USB mics.

21ST CENTURY BREAKDOWN: 1 triple
Section: Chorus 1
Phrase: "To the 20th century deadline. Ooh."
Method: With one person on blue and one on brown, sing "To the 20th century deadline" then jump up to yellow for the long "Ooh" sustain.

THE STATIC AGE: 4 triples
Section: Chorus 4
Phrases: 2x of "I can't see a thing in the video (Whoa). I can't hear a sound on the radio (Whoa)."
Method: With one person on blue and one on brown, sing "I can't see a thing in the vid-e", then quickly jump up to yellow for the short "Whoa". Repeat for "radio" phrase, then repeat both again.

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EDIT: This one is a little more complex, but can be done with 2 people and 3 mics.

BEFORE THE LOBOTOMY: 6 triples
Section: Bridge
Phrases:
(Too late.) We're lost like refugees.
(Too late.) Like refugees.
(Too late.) We're lost like refugees.
(Too late.) The brutality (Ah)
(Ah) of reality
(Ah) is the freedom that keeps me from.

Method: Give 2 mics to the blue person and 1 to brown. For the first 4 phrases, stick to your own color, and brown sustains the same note to fill the yellow "too late" parts. For the last 3 phrases, brown stays with brown, and blue concentrates on yellow. There's enough overlap that the same note into 2 mics will fill both blue and yellow. (Works best on Easy, but doable on Expert.)
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You can view/practice the specific sections via Training, Practice Mode. Select the song & difficulty, then a start & stop section (can be the same). LT toggles which part(s) display onscreen, and RT toggles a slightly louder pitch guide for a specific part. (The pitch guide sometimes turns off after each restart.)

Note: The triple(s) must be hit in Quickplay or Career, not in Practice Mode, and the achievement pops at the results screen.

You can get this BY YOURSELF with 2 USB MICS!

Play the song "Static Age" on easy difficulty. The goal is to sing IN BETWEEN THE PITCHES!! There are only 4 Triple Harmonies the entire song, so just follow the video ;D

I was actually able to get this by myself using three microphones, on the song "Whatsername." There are only five triple harmonies on this song, and most of them overlap. It took me a few tries, but it is doable. Just make sure you fill up both overlapping bars. I found that if I just moved my voice up-and-down to cover all of the bars that they seemed to fill up faster, rather than trying to fill up one bar and then move to the next bar before the section ended.

Like I said, it may take a few tries, but if you can't get it by yourself, you can definitely do it with two people rather than trying to find three. Also please note, you will need three microphones to get this achievement.

The Song i have done it on is Static Age towards the end of the song there are 4 triple harmonies and on easy i hit the double at the some time with 2 mic and on the 3rd mic you just say whoo-oh

Only hard part is having 3 mics which is self explaintory

This was done solo and this is where the idea came from
Solution for Triple Awesome In Rock Band 3

For those doing this achievement with three players and no DLC, I recommend Welcome to Paradise. The triples in this song are exactly the same on each pass - one Ohhhh held slightly higher than the lead vocal, and a second Ohhhh that descends from the high line to overlap with the lead vocal. We did this on Medium on the first try with two players who had never one vocal harmonies before, and I expect (but cannot guarantee) that on easy you would be able to trigger it with two players sharing three microphones.

If you can't find another person to play harmonies with you but you have 2 mics, you could always plug one mic into your computer, download an audio editor like Audacity, and record yourself playing a song; 21st Century Breakdown is easiest, like Ripley suggested. Hold both mics up to your mouth and play the song in practice while you record with another mic - playing the song while recording is a lot easier than just straight recording since you can tell you're actually hitting everything. Then you just have to play back the recording into one mic and sing with the other. The hardest part is syncing everything up; I found it easiest to record the blue and yellow parts, then sing the orange part myself. Since you have the blue recorded, you can tell from only the first phrase whether or not it's synced correctly instead of fiddling with a harmony late in the song.

Also, since any difficulty counts for the achievement, definitely play on easy.