Since unarmed strikes are considered melee weapon attacks, and the 'smite' spells don't have the same restriction as booming blade or green flame blade (the part where it requires "a weapon" in material components)

Would you be able to cast and trigger the 'smite' spells (like thunderous smite, searing smite, etc.) with unarmed strikes as they are melee weapon attacks?

You need a weapon for all the smite spells (except wrathful smite)

Smite spells require a weapon

Thunderous smite says:

The first time you hit with a melee weapon attack during the spell's duration, your weapon rings with thunder ...

In fact, every smite spell has similar language specifically calling out the need for a weapon (except for wrathful smite see below). By a strict RAW reading though the weapon does not actually need to be the one that makes the attack, but you still need a weapon.

Regardless, I think it is pretty clear here that the spell is intended to utilize the weapon that you use to make the melee weapon attack even if the wording leaves a bit of a loophole. Either way, the spells do require a weapon.

The body/unarmed strikes are not weapons

The PHB says:

Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). (PHB 195)

Suggesting that though body parts are not weapons, the unarmed strike still counts as a melee weapon attack but explicitly not a weapon.

And the Sage Advice Compendium clarifies this and makes it explicit:

[A]n unarmed strike counts as a melee weapon attack, even though the attacker’s body isn’t considered a weapon.

Since the smite spells specifically call out a weapon in their description, that is what they require. And since unarmed strikes are not considered to be weapons, then they do not qualify.

Wrathful smite does not need a weapon

Wrathful smite is, for some reason, the only smite spell that does not specifically call out the need for a weapon.¹

The first time you hit with a melee weapon attack during the spell's duration, your attack deals an extra 1d6 psychic damage.

Thus, since the unarmed strikes do count as melee weapon attacks, a character would be able to use an unarmed strike to cause the effect, no weapon required at all.

^{1 - Thanks @Someone_Evil for catching this.}

I agree with Rubiksmoose's answer that the smite spells (except wrathful smite) require a weapon.

However, I thought it might be useful to share my own house ruling on the subject. I decreed that, while the resident paladin couldn't use a smite barehanded, a gauntlet could count as a weapon for the purposes of the various smite spells.

Note that, to the best of my knowledge, gauntlets arent considered weapons anywhere in the RAW. They do, however, provide a way to "disarm" a paladin so that they can't just punch holes through monsters with impunity. I felt it wouldn't break the game too much to tweak the rules in such a fashion, and I honestly wanted to see where she was going with it.

The paladin eventually took advantage of this ruling to literally backhand a wereboar to death. It was a very cool moment in-game, and I regret nothing. As always, Your Mileage May Vary.