Speaking frmo personal experiance, i had been married for 6 years and been with my partner for 14 years.
I never ever checked his phone or email, never even thought of it.
But in a way i wish i did now, reason being is he had been having an affair and it started off over texts.
How i found out is when i suspected something he admitted to one incident, but then i knew somethign wasn't right, for the first time ever i went on line and checked his mobile account which i guessed his password - ironically it was my name!
Then to my horror fuond the whole truth.
Putting that aside we are now going through a divorce and i am now with a newish partner, been togther for 1 1/2 years.
I now find checking his texts and email a really bad habit, - and before any one goes mad saying i shouldnt tar him with same brush - he cheated on me last year with his ex - so this is mainly why.
Im not saying its in no way right doing it as i woudlnt have before, but what i would say is if you do suspect something i dont see harm as if he wasn;t up to something then there would be nothing to hide.
I would also say if you didnt do it as you had suspicions then stop now as you may ruin your relationship over doing it, as he may find out.