I think the first and most important question is; does he have reason to cheat? How happy are you together? I ask because if one (and thus both) of you are unhappy, it doesn't matter whether he's cheating or not.
Based just on the very basic information in the OP, I would say that your marriage is in trouble. If you don't even trust your husband enough to ask him what he's up to and believe his response, then I think that's a marriage-killing issue. Even if he's not unfaithful, the aura of suspicion can slowly kill all the good things you experience together.