When features require concentration or end when falling unconscious, this is explicitly stated
Let's compare the Rune Knight's Runes to the Ring of Shooting Stars, the Draconic Sorcerer's Draconic Presence feature, and the Quasit's Invisibility action:
[...] The spheres last as long as you concentrate (as if concentrating on a spell), up to 1 minute. [...]
[...] For 1 minute or until you lose your concentration (as if you were casting a concentration spell) [...]
The quasit magically turns invisible until it attacks or uses Scare, or until its concentration ends (as if concentrating on a spell). [...]
These all explicitly state that they require concentration. But when looking at the Rune Knight, no part of its description states this; therefore, runes do not require concentration.
As far as falling unconscious is concerned, when effects have specific end conditions, those are stated. We can see this in the Barbarian's Rage, the Druid's Wild Shape, the Oath of Devotion Paladin's Channel Divinity: Sacred Weapon, and the Storm Rune itself:
[...] Your rage lasts for 1 minute. It ends early if you are knocked unconscious [...]
[...] You automatically revert if you fall unconscious, drop to 0 hit points, or die. [...]
[...] If you are no longer holding or carrying this weapon, or if you fall unconscious, this effect ends.
[...] enter a prophetic state for 1 minute or until you're incapacitated. [...]
These features all explicitly end when unconscious (and in the case of the Storm Rune, incapacitating alone is enough). Since all the other runes lack any similar text, we can conclude that they do not end when falling unconscious.