I don't know that you'd actually have to take your hand off the weapon. Unarmed strike can encompass kicks, elbows, headbutts, etc. can it not?

As it is considered a weapon, making an unarmed strike is equivalent of making a one-handed sword attack.

@DaleM part of the addendum is asking about two-handed weapons which would not be Monk weapons and would be subject to non-monk weapon rules (thus the usage of str), I've edited the answer a bit.

NRBH but I can't see a) why the 2 handed weapon needs to be stowed or b) why the Dex cannot be used. For a) FoB allows two extra unarmed attacks when you take the Attack action - it does not say that the original attack must be unarmed or with a monk weapon (Martial Arts does). For b) Martial Arts allows Dex when using unarmed or wielding (using) monk weapons - it doesn't say you can't be holding a greatsword at the same time.

@DaleM a: unarmed attack is a 1 handed weapon. You have to a hand free to make it, no matter what body part you are using. So you have to stow a two hander. b: Monks can only use Dex for Monk weapons (or finesse weapons), there are no two-handed monk or finesse weapons. A two handed weapon can't be a monk weapon (Explicit exclusion), and there aren't any two handed finesse weapons right now.

To add to the previous comment, that is why the quarterstaff is so popular among Monks; it is a monk weapon that can be used with two hands (making it a d8 attack), plus it allows for the additional Martial Arts' unarmed strike

Sorry, I must continue to disagree - where is the definition of 1 handed weapon? There is a definition of 2 handed but simply because a weapon isn't the latter does not automatically make it the former. This is left to DM judgement and while in most cases it is obvious that a weapon needs to be held in a hand this is not the case for unarmed strike. If the monk (or anyone else) wants to kick, knee, elbow or head-butt then that is an unarmed strike and it is not "1 handed".

@DaleM is there a reason to believe it doesn't require a free hand? I mean, does a Mace? since it has the exact same properties except for the damage die. At this point you probably need to add your own answer making that argument. Or we can continue the discussion in chat: chat.stackexchange.com/rooms/15929/dd-5e-discussion-goes-her‌​e

This answer requires an update due to the changes in the errata which have since clarified that unarmed strikes are not weapons. This negates the need for a "free hand" in order to utilize an unarmed strike.

Ditto LFC ^^. Now that unarmed strikes aren't weapon attacks (nor on the weapon table) and their proficiency is separately called out, this answer reads very strangely.

At no point does the PHB indicate that unarmed strikes are one-handed or that they're simple weapons. It states clearly that you can use any part of your body and that every character is proficient with them. Yes, even wizards.

@jack the original phb text has unarmed strikes in the weapon table under simple weapons, they do not have the 2-hand or martial properties so they are considered 1 handed and simple. However, since this answer was added, errata has been published removing unarmed strikes from said table

Unarmed strike has been taken off the 'weapon's list' in the recent errata. unarmed strike is no longer considered a 'weapon' attack, unless a specific rules calls for it (ex.: monk's unarmed attacks are weapon attacks)

@KilrathiSly: That is also inaccurate. Unarmed strikes were removed from the weapons table in the very first errata in 2015 (not a recent one), but they are always considered "melee weapon attacks" even if they do not use a weapon.

Also, wax eagle's profile says "Last seen Feb 25 '17 at 2:12", so I wouldn't expect this answer to be updated anytime soon...