I have the cheonjae textbooks and we just skip the chants/songs. If we really want them to sing something, we just find a pop song to teach them.

Hi

My short reponse has gotten longer than I expected.  This is what happens at my school and seems to work well (mostly).  Maybe it helps that I've been here for a few years and the grade 6s just accept it as the way I do things.

We use the Cheonjae books for grade 3-6 and the teachers work out some actions which we do.  (we practise them beforehand to make sure we remember them and that they fit the timing etc).  We don't usually use the actions from the book as they don't always fit well.

We use the same actions for when we say the sentences when learning them.  TPR (Total Physical Response) can be effective - even for the grade 6s.  If the actions link well to the words then it can help a student to recall the words in English.  If a student is struggling to know what to say I do the action and frequently that helps them to recall the English - this is especially satisfying for the lower students.

If you know some sign language (or can find it online) you could use those hand signals as the actions - and students might be keen to know it is a real language they are speaking with their hands.  It would be ideal if it was Korean sign language as that could be useful in Korea for Koreans.

The students will not enjoy or participate in the songs and chants if the teacher doesn't.  Teachers are actors! so you can 'act' as if this chant/song is great/exciting and worth doing!  (Who knows you might find yourself enjoying some of them!)  I do agree that some of them are not good at all and that the phrasing/timing can be a bit(!) awkward at times.

Singing and chanting are excellent ways to learn and retain language so I don't think they should be eliminated.  You can find substitute songs on youtube but try to make sure they are simple enough for the students to sing fairly easily.  I find Pop songs may be interesting for the students but are they too hard/complex for most of the students?  Are they able to sing along (and understand what they are singing - I can sing along in Korean but I don't know what I am singing and it doesn't help reinforce the language).

How are they taught in the classes at my school?
1) We play the song through once with no words visible but with me (and sometimes the co-teacher) doing the actions (and usually lipsynching).
2) Then we do the 'listen and repeat' singing line by line (some times 2 or 3 times if a particular line is tricky) while doing the actions. 
3) Then we sing the song through 2 or 3 times. 
4) The final time we sing we award group points (eg 3 points if all students are singing and doing the actions - I accept it if a low student is so focussed on trying to say the words that the actions are minimal).
5) We tend to start the next few periods with the chant and/or song for the rest of the unit.

I'm not saying that all classes sing enthusiastically but more often that not it is not too bad.  The key seems to be (at my school) that if the teacher is having fun the students will find it more enjoyable too.

Enjoy your singing

I've been taking my guitar into class and adapting the chant to some simple chords. It's pretty fun for me too since I try to fit each chant into a different sort of genre. If you or the school happens to have a guitar it's very simple to learn a few chords if you can't play yet. It doesn't require a lot of guitar skilz to do a chant.

One thing you can do with your sixth graders -- if they're good -- is to have them pair chant as a sentence stress activity. I never use book chants: the Cheonjae books lack them at 5th and 6th and for 4th its just a second, even poorer song. Rather, elicit the target language, scrawl it up, grab a pen and tap it on a metal surface. Draw attention to the stressed syllables and then have half the class repeat the stressed syllables over and over as you tap a pen, then get the others to read the whole sentence so the two groups are in sync, slowly increasing the tempo as a challenge. (Swap roles later.)

It's a bit tricky to pull off, but they'll appreciate it as a functional learning experience whereas the pure book chants tend to come across as a bit kiddy and this naturally will rub them up the wrong way.

Regarding book songs, my first coteacher down in Daejeon would largely play them at the start of the lesson or during the tidying up phase at the end of a game. Her approach is the best I've encountered. Play them for every lesson in the unit, from the get go. Students come in, and sit down quietly whilst subtitled pop songs or interesting videos play at low to mid volume. Then, when its time to start, we switch the to the book song. From the second lesson onwards in a given unit, they're expected to try to sing along.

One playthrough and the lesson begins. It's a functional classroom management tool and we don't dwell. If you can get your coteacher to help explain why we use the songs (sentence stress, rhythm, to aid with revision -- the song lyrics are often the only full sentences in their books -- and plain ol' earworms), that helps immensely, but really, the worst thing you can do is patronise the kids by setting aside a whole chunk of lesson to them. Also, be prepared to bite your tongue. If you let contempt for the songs (yes, they're crap, we know) show, even through subtle body language, they'll pick up on it immediately and then both you and whoever teaches these kids next will be cast adrift.