From what I can gather, most music besides some specific genres is based on top of a background beat layer. I happen to be one of the (apparently) few people with a strong dislike of any kind of repetitive beat (I find it painful and too occlusive of the actual melody), and have trouble finding music without it.
When searching for music, I've used the terms "beatless", "no beat(s)", and similar variations with very little success.
Do you know any specific qualifier one could use to search for music with strictly no percussion ? Not even light, quiet percussion.
If you know about specific genres that are characterized by their absence of beats, I'm also interested. Especially electronic music.
EDIT : the "ambient" and "drone" genres have already been mentioned below