[Update:] With reference to Noam's comments below, here are some data for the relative frequencies of the sum of three consecutive primes being divisible by the first four odd primes. These are averaged over samples of $400,000$ primes beginning at powers of ten, which are given in the first column; note that these refer to the numbers $x$ themselves, not the indices $n$ of the primes.
I also looked at the joint distribution of the residues modulo $3$ for the three primes. There's a significant preference for avoiding repeated residues; for instance, at $x=10^9$, the repeating patterns $1,1,1$ and $2,2,2$ have relative frequencies around $0.095$, the alternating patterns $1,2,1$ and $2,1,2$ have relative frequencies around $0.150$, and the remaining mixed patterns have relative frequencies around $0.128$, which is almost completely explained by $1,1$ and $2,2$ having relative frequencies $0.445$ and $1,2$ and $2,1$ having relative frequencies $0.555$. I'm trying to work out a probabilistic model for these effects.