When I travel, I always wear the neck-style RSteves-brand wallet under my t -shirt/blouse/jacket whatever..depending on seasonal attire. Obviously early spring/fall/winter travel is easier to "hide" it. I pretty much always wear a scarf tied around my neck to "hide" the bulge of the neck wallet. For summer, select some very light-weight cotton scarves and tie them loosely with the knot just above where the money-wallet would fall, if that makes sense. Otherwise, yep, it is one big crunchy square that stands out like "what is that?" on my chest...not at all attractive. Often I will slip the waist-style RSteves money belt in our luggage, just so I have that as an option, if there is a situation where I do not want to wear the neck one. But, I always use an safety pin, in addition to the clasp, just to avoid the highly-unlikely situation of the clasp working itself open..
My husband wears his holster style (under his arm). Does it get wet and gross sometimes?.......oh gosh, yes....especially when temps are in the high 80s and above......yuck. How do we deal with that, we take two or three of them (if it is hot weather) and switch his stuff to a fresh one (while the other is being washed). We also use a very light-weight baggy as an extra double moisture protection for the passport, etc. For his, I have adjusted the cord length "just so" to fit around his neck/under his arm, so he does not have to deal with the plastic cord/string adjusters driving him crazy while he wears them.
Oh, and to further explain a bit, what goes into my spouse's under-arm storage pretty much NEVER comes out except in our hotel room. His is the "deep storage." If he will need his passport on certain days (like flight days), then he wears a second neck wallet that slips down under his cotton, dress style, button-up shirt (he does not wear knit shirts) in the front, where he places his passport in the backpocket of the neck-wallet for ease in accessing it .
Try a few things, and you will determine what you like best.
If we are in a "protected-type" situation (private dinner in a super safe location or (rarely..like on a small ship..because the purser keeps the passports on those and those typically have cabin safes), there are times I will stick my neck wallet in a zipped inner pocket of my travel purse, which is always then worn cross-body (so it is not set down anyway...always on "my person." But, those situations are rare.