Asikis was the third child of Victor Asikis of Lesbos and Marianthi from the Kontoskali district of Constantinople. Ever since he was a child, he thought he had an exceptional voice and bought an outi at the age of 15. He sang and played the outi for family and friends but did not begin his professional music career until after the Asia Minor Disaster. In 1922, after the Asia Minor Disaster, Asikis and his wife, Eirene Sotiriadou, and two children, Thanasis and Victor, moved to Athens, Greece. They moved to the Vyronas section of Athens and had four more children: Anastasia, Nikolaos, Basiles, and Manolis.
Having moved to Athens, Asikis tried to continue his professional career of fabricating bronze bed frames, but failed after six months. He would frequent a club called "The Asia Minor", the center of all laika at the time, where he was warmly accepted by contemporary musicians. In the late 1920s, he recorded music for Columbia, Odeon, and Parlophone as did other musicians at the time, like Kostas Skarvelis.
In total, Grigoris Asikis recorded 82 of his songs. In the 1930s his compositions were sung by many famous artists of the time including Kostas Nouros, Roza Eskenazi, Stellakis Perpiniadis, Rita Abatzi, Marika Politissa, Kostas Roukounas, Kostas Tsanakos, Marika Kanaropoulou, and Georgia Mettake. In the later part of the decade, he abandoned composing songs with Asia Minor instruments, i.e. outi, lyre, and violin, and wrote songs using the bouzouki. Some of these songs were sung by famous artists and close friends of his Markos Vamvakaris, Yiannis Papaioannou, and Apostolos Chatzichristos.
Moor's own background is rooted in the post-punk Britain of the 1980s, but in more recent years he has collaborated with many musicians from varied backgrounds and disciplines. Many of these collaborations are duos, with amongst others Cypriot composer Yannis Kyriakides with whom he performs a set of rebetika songs (urban Greek tunes from the 1920s and 1930s), French sound poet Anne James Chaton, ex bass player from Dog Faced Hermans Colin Mclean, who now works with electronic music and live sampling, Brooklyn-based DJ, producer and writer DJ /rupture, and Lebanese Paris-based saxophonist Christine Abdelnour.
Moor has also begun composing soundtracks for films, working former Dog Faced Hermans vocalist-turned filmmaker Marion Coutts, with Iranian filmmaker Bani Khoshnoudi and US experimental filmmaker Jem Cohen. His latest projects include a quartet with Ken Vandermark, Terrie Ex and Paal Nilssen-Love called Lean Left, a trio with Berlin-based artists and musician Alva Noto and Anne James Chaton called Decade, and a trio project with Anne James Chaton and Thurston Moore called Heretics. In 2013 he began a duo project with John Butcher. Moor lives in Amsterdam at present and continues to be a full-time member of The Ex. He is a founding member of the music label Unsounds, in Amsterdam, with Yannis Kyriakides and Isabelle Vigier.