Regular expression for $\{x : \#_{01}(x) \mod 3 = 0\}$
Note that any amount of ones at the start or zeroes at the end has no influence on the number of $01$ pairs. Then in the middle you want any multiple of three of $01$ pairs, giving:

$$1^*(0^+1^+0^+1^+0^+1^+)^*0^*$$