Welcome back to the game! I'd like to answer your question in two different parts: the mechanics and the ascetics.
I think there is a core misconception here:
balanced against typical adventure tropes of killing monsters, taking all their stuff and becoming a wandering one-man magical army.
which I think comes from your background with 1e and 2e.
5e has a much lower dependence on magic items. In particular, I invite you to take a look at the table Starting Equipment on page 38 of the DMG. In case you don't have it handy, it breaks down expected starting wealth into four character level ranges (1-4, 5-10, 11-16, and 17-20) and maps them onto Low, Standard, and High magic campaigns. The recommended number of magic items for the most magic-y, high powered game, a level 20 high magic campaign?
Three Uncommon, two Rare, and one Very Rare item. The main reason you want magic items in the first place is to give yourself new abilities (teleportation, fire immunity, etc) or improve existing ones (hit chance, spells per day, etc). In 5e, neither of these is as much a problem as in previous editions.
For the first, classes are much more flexible than previously. Only two classes don't have access to spells (Barbarian and Monk), and even these have very spell-like options (speaking with nature itself, or using your mind to light your magic fists on fire).
For the second, 5e has a concept called Bounded Accuracy. At first level, a fighter's to-hit might be +4. At max level, it could be around +11. Mirroring this, the AC for creatures goes from around 13 on a standard Orc up to 22 for an Ancient Red Dragon or 25 on the extremely defensive Tarrasque. Since the numbers don't really vary much, you don't need much in the way of items to keep up with average monster defenses. The damage bonuses (usually in the form of more damage die) get bigger, but the real meat comes from multiple attacks rather than stupidly big numbers.
So, mechanically speaking, keeping your magic items to a minimum won't impact your game that drastically, so long as your DM isn't giving them out like candy to your other party members.
Magic items in 5e are all so valuable they're literally priceless. Price ranges can be found in the DMG, but it's advised to trade services and favors, rather than outright selling things. This can be used in your favor. Willingness to trade away your magic items lets you trade direct combat ability for greater influence (compared to your compatriots) in the social environment of your campaign. This is likely to vary greatly on your DM and game style. Great for city sleuthing, poor for straight dungeon crawls. There are more official, but optional, rules for this in the form of Organizations on page DMG 21. Talk to your DM about setting up your monastic order as an organization in the world.
A roleplaying alternative to straight giving or trading away your valuables is to consider one or two of them as holy or otherwise special relics. Your character may well consider a particular item or monk weapon to be special to his order, and it's his duty to see the item used in some way in the world. Of course, your monk may decide to keep an item or two as a reminder or keepsake of a particular adventure, relationship, or lesson learned.
Lastly, and perhaps most on point, a few of the rarer magic items are effectively consumables that give permanent benefits. Take a look at the Tome of Understanding in particular. You read it, learn from it, improve yourself, then rid yourself of the burden of possessing it. Depending on this approach will require something of an accommodating DM, but note the first paragraph in this section. Long-lived or immortal creatures such as dragons may well be willing to let you peruse their libraries for a week at the cost of some bauble you found. Permanent possession of a new item could easily be worth effectively a 100-year loan of a different one.