"Cinematic" for Video Games
Solution 1:
You are probably looking for ludic. Wikipedia gives the definition:
The adjective ludic originates from the Latin noun ludus, meaning "play, game, sport, pastime."
Related terms: ludology, ludological, ludography.
Of course, these words are rare, and may not be understood by a wide audience. Some more widely understood terms would be game-like (may or may not be hyphenated) or game-y.
Solution 2:
Oh, I love video games! Old video games sound the way they do because they use sound synthesizers. Doom in particular makes a very good example because the music was meant to be played on the Roland SC 55 Midi Synthesizer if I recall correctly, and there are various home console ports that would have used lower quality integrated synthesizers, like the S.N.E.S. which used the custom Nintendo S-SMP soundchip and and the Sega 32x which if I recall correctly would have utilized the Sega Genesis's Yamaha YM2612 F.M. synthesizer.
As such the obvious answer to this question is synthetic:
SYNTHET'IC
SYNTHET'ICAL, adjective Pertaining to synthesis; consisting in synthesis or composition; as the synthetic method of reasoning, as opposed to the analytical.
The American Dictionary of the English Language by Noah Webster
The only problem is that sound synthesizers aren't always used for video games, and when applied to music it's sometimes taken as a criticism. As such, you might want to use the phrase synthetic timbre. Timbre is a word meaning:
The quality of a sound independent of its pitch and volume.
Excerpt from the Wikitionary definition of Timbre, licensed under the CC-BY-SA license.
Wikipedia even cites the words "synthetic timbre" on their Electronic Music page as being used in Sounding Art: Eight Literary Excursions through Electronic Music, Aldershot: Ashgate Publishing, Ltd, ISBN 0-7546-0426-8. Also before you read it, I should disclose that my pseudonym is a misnomer. I am not a musician by either trade or hobby.
[Sidebar: Right now I feel my citations are a little weak, which is because I'm busy. I'll fix them up later.]
Solution 3:
What's a word suffix that means "of or relating to" ?
video-game-esque
-esque
suffix
(forming adjectives) in the style of; resembling.
"It sounds very video-game-esque."