There is no flanking rule as a default; there are, however, a couple of optional rules proposed in the DM’s handbook. It should also be noted that these are part of a general set of rules dedicated to playing using miniatures and tactical maps and even then they are marked as optional.

The flanking rule states:

Flanking on Squares. When a creature and at least one of its allies are adjacent to an enemy and on opposite sides or corners of the enemy’s space, they flank that enemy, and each of them has advantage on melee attack rolls against that enemy.

When in doubt about whether two creatures flank an enemy on a grid, trace an imaginary line between the centers of the creatures’ spaces. If the line passes through opposite sides or corners of the enemy’s space, the enemy is flanked.

Flanking on Hexes. When a creature and at least one of its allies are adjacent to an enemy and on opposite sides of the enemy’s space, they flank that enemy, and each of them has advantage on attack rolls against that enemy. On hexes, count around the enemy from one creature to its ally. Against a Medium or smaller creature, the allies flank if there are 2 hexes between them. Against a Large creature, the allies flank if there are 4 hexes between them. Against a Huge creature, they must have 5 hexes between them. Against a Gargantuan creature, they must have at least 6 hexes between them.

DM’s Handbook p.251

An alternative optional rule offered by the DM’s guide is called facing.

If you want the precision of knowing which way a creature is facing, consider using this optional rule.

Whenever a creature ends its move, it can change its facing. Each creature has a front arc (the direction it faces), left and right side arcs, and a rear arc. A creature can also change its facing as a reaction when any other creature moves.

DM’s Handbook p.252

This rule suggests the following effects:

An attacker in a creature’s rear arc has advantage
Shield bonuses only apply AC to the front arc and side arc of the shield arm.
Some creatures will not have limited arcs, like an amorphous blob

This is arguably more complex to adjudicate and manage than flanking is.

A potential issue one could take with flanking in 5e is related to the change in Attack of Opportunity rules. Previously, circling a creature tightly, even with a 5ft reach would provoke an AoO, as you were moving through threatened squares. In order to flank and avoid an AoO you would typically need to go wide or use an ability. In 5e an opportunity attack is only provoked when you move out of a creature’s reach without disengaging. In my opinion this makes it effectively trivial to flank.

As a further counter argument, keep in mind that DND 5e adds the “help” action, which can easily be used as an analogous — yet more general and less powerful — mechanic.

Alternatively, you can aid a friendly creature in attacking a creature within 5 feet of you. You feint, distract the target, or in some other way team up to make your ally’s attack more effective. If your ally attacks the target before your next turn, the first attack roll is made with advantage.

Player’s Handbook p.192

In order to create an advantage, one attacker must focus on helping via a distraction or combat engagement at the cost of not being able to effectively score a hit. However, this still gives the other attacker a far better chance at scoring a hit. This makes this more useful against powerful opponents while being unimportant against lesser ones. I feel this makes this a much more tactical choice than mere positioning, while still providing a simple and generic mechanic that feels balanced and is unlikely to be overused.