I just did a squinty through my books. Most of the old pics show the dirk on the right side. Quite a few are on the left. Right makes sense since the dirk would be held opposite the sword hand which is typically the right. Go with the cross draw idea.

Some people carried several dirks, or dirks and bayonets, so, really, go with what works for you.
If anybody challenges you, make them explain it. You've already got the answers here.