Not only wire. May need an increase in raceway and larger boxes. Depends on your branch circuit layout. Your CCC in a raceway increases as well. For 3, 120 volt circuits that would be 6 CCC's. You can see how 6 MWBC versus 6 line to neutral Branch Circuits will increase wire size and race way size or increase the number of raceways for the same conductor size.
Depends on you lay out the home runs.
An example would be:
MWBC, 2 sets, netural don't count.#12 home runs.
6 #12 CCC, derate for 6, #12 @30 amp with derate to 80% = 24 amps. One 3/4" home run.
Line to neutral home runs. 6 circuits,12 CCC
#12 home runs.
Derate to 50% or 15 amps. Now we have to change quantity or wire size.
Using #12 would be 4 circuits 8 CCC per raceway.
Now we're back to 9 #12 in that 3/4 home run.
Or we change to #10 then your back to 6 circuits and 12 CCC. #10@40 amp derate 50% or 20 amps.
Now you 13 #10 in a 3/4 emt. No you have to bump raceway size to an 1".
I used a wire type EGC above.
For box fill each additional #12 will require a 2.25 in³ and #10 bumps you to 2.5 in³. For line and neutral. The EGC fill depends on what code year.
A, 4 square 2-1/8 is good for 13 #12 and 12 #10
A, 4 square extension is good for 9 #12 and 8 #10
A, 4-11 2-1/8 is good for 18 #12 and 16 #10
I think you get the point on fill.
Just something to consider based on how you fill what raceway. There is a trickle down effect after the home run.