Personally, in 12 extended trips to greece since 1999, I have stayed in more than 60 hotels, and have Never, repeat NEVER, had any incident in hotels with anyone touching any of my possessions. I do keep my paperwork/passport extra money in a flat "hidden pocket on the "floor" of my suitcase and lock the suitcase with a cheap CVS lock ... but for years just stuck it in a bag side pocket & left bag unlocked. I dont stay in big international chain hotels that may hire foreigners, mainly family-owned places run by Greeks. As I repeatedly say on these forums, "Petty THeft is NOT a part of Greek Culture" (taught me by an American,married to a Greek, who brought up her child in both countries -- she also said, Greek moguls may steal one another's shipping fleet, but everyday Greeks do not stoop to petty theft). So I feel safe going about with only my day's money needs, in a change purse.
When I travel by plane to/from Greece, or taking a long ferry w. all possessions, I do carry those items (Passport,ticket cc & cash) on my person --- but like most women I don't care for money belts... why add bulk around one's WAIST?!?!?!? Instead, I have a small fabric "neck pouch" - loops around my neck, hangs in front under my shirt. Available at any travel-accessories area, in dept store.