So much music today relies heavily on drums as their primary rhythmic motivator. But it doesn't have to be that way and in fact it may help you carve out a musical niche to downplay their significance in your music. If you want to explore music that does not rely on drums as a driving force I suggest you delve into the world of classical art music. There are some massive rhythmic symphonic and ensemble works that feature no percussion or simple timpani underscoring the low strings.
- Listen to how the strings provide a driving rhythm for themselves.
- Notice the timpani (big, deep drum) playing along with the low strings along with sustained strings in the melodic lines.
I gather it may not be your style but you can still learn from it.
You can also look at other sounds or instruments as "beat-makers". Pianos are percussion instruments. You'd be fascinated at the sounds you can get from a string section (
http://youtu.be/sqrZSYzg1GU). Also rhythmic bass guitar riffs. Arpeggiated synth lines. Etc.
Also, you can easily have a primarily slow ambient texture in a song and lace it with subtle driving rhythms (quite bass in a faster time signature, background arpeggios, off-beat accents in the accompaniment) which will keep the piece moving along yet in a relaxing way.
I'm not sure if this is quite what you are looking for (and now I just saw that you play percussion in an orchestra) but perhaps it helps.