About a week or so ago I had this idea for non-traditional monk traditions for 5e. Basically, the idea was to have characters who fought bare-handed like monks, but had slightly less regimented backgrounds, and consequently, less "Far Eastern" abilities.
I initially came up with three traditions: the Brawler, the Wrestler, and the Kensai. I have since expanded the idea into a whole new class, based mainly on the Monk from the PHB, but with some distinct differences. And in the process, I decided that the Kensai was actually more like a very focused traditional Monk.
So, here is the Kensai Tradition. It isn't playtested, so you may want to tweak it a bit here and there. But the idea is that this is a fourth Tradition your Monk character could choose from at 3rd level. I've also included a few magic items that are geared specifically for traditional Monk characters.
Also, as usually happens when I get a bug like this up my ass, someone else has already done the Kensai as a full class.
Have a look here. It looks pretty good, honestly.
As always, comments and criticisms are welcome.

Kensai
There are warriors who gain total mastery
of a specific type of weapon, and then there are the kensai. A kensai is a monk who has dedicated
practically his entire existence to perfecting his ability with a specific type
of weapon, to the point of it being a supernatural connection. Although a kensai can use other, simple
weapons, if there is a weapon of his chosen type available, even if it is
inferior to other available weapons, he will always choose that one. At 1st level, the kensai declares
his chosen weapon, and he will use this weapon (or another one just like it)
until the end of his days. This weapon
can be simple or martial, but cannot be a ranged weapon. Eventually the kensai will focus his efforts
with a specific weapon of this type, often specially commissioned, and always
of masterwork quality. However, a kensai
will never use a magic version of his
chosen weapon.
Special note: Any instance
where a monk would gain an additional unarmed strike, the kensai can be assumed
to make that additional attack with his chosen weapon.
Perfect Form
When
taking the kensai tradition at 3rd level, you improve the die-type
of your chosen weapon by one step. From
then on, any weapon of this type you use rolls this new die for damage. For instance, if you choose the longsword,
any longsword attacks you hit with will deal 1d10 damage, instead of 1d8. If you choose a greatsword, any attack will
deal 2d8 damage, instead of 2d6.
Ki Weapon
At
6th level you can channel your Ki into your weapon, per the monk’s Ki Empowered Strikes feature. However, when using your chosen weapon, you
also add one half of your Proficiency bonus (rounded up) as a magical bonus to
attack and damage.
Soul Weapon
At
11th level, you must choose a specific weapon to focus on. This weapon becomes soul bound to you; you
can never be disarmed, and you will never fumble with this specific
weapon. Additionally, you score a
critical hit on a natural roll of 19 or 20 with this weapon.
My Weapon Is My Life
At
17th level your Soul Weapon can be summoned from a distance, and
will magically appear in your hand instantly, so long as it is on the same
plane of existence. Additionally, you
can spend a Ki point once per round to make a single ranged attack with the
weapon, attacking any target within 50 feet of you.
Magic Items for Monks
Monks
are not above the use of magic items, though their normal restrictions on
weapons and armor apply. Items such as Rings of Protection, Bracers of Defense, and Girdles of Strength are fairly common
for monk characters to use when available.
Below
are four magic items that are specifically intended for monks.
Gauntlets of the Iron Fists
These
hand coverings are usually in the form of wraps of sturdy cloth or leather, and
radiate moderate magic. Wrapping one’s
hands with these gives the monk a magical bonus of +1 to +3 to hit and damage
when making an unarmed strike, depending on the power of the enchantment.
Intercepting Staff
This
+1 staff is made from dark cherry wood, and radiates mild magic. While wielding the staff, a monk may expend a
Ki point to activate the staff’s Interception ability for that round. When this is used, the monk makes an attack
roll, the result of which becomes his AC for a single opponent’s next attack. If the opponent misses, and the monk’s AC
roll was higher than the opponent’s AC, the monk strikes with the staff as a
free action, automatically dealing 1d8+1 (plus Strength modifier) in damage.
Ki Medallion
This
medallion is a small disk or amulet on a golden chain worn around the monk’s
neck, and radiates faint magic. By
meditating with the medallion in hand for one hour, the monk can suffuse the
medallion with Ki points equal to half of his current maximum rounded
up. These Ki points can be spent just
like the monk’s own Ki points.
Ring of Physical Perfection
This
plain, onyx ring radiates moderate magic.
A monk wearing such a ring may expend up to five Ki points to heal
himself as a full action. For each Ki expended the monk regains 1d8 hit points up to his maximum. Once all five Ki points have been used, the
monk must meditate on the ring, regaining one Ki point of healing for every
half hour of meditation, up to a maximum of five (2.5 hours).