There have been many anime composers over the years, and while anime soundtracks are big business in Japan, there have been few notable, long-term composers of anime music before the 2000s.
Another early, notable anime composer was Shigeaki Saegusa, composer for Mobile Suit Zeta Gundam in 1985. He was a classical composer who produced a symphonic score for this series, and the series went on to be extremely popular (one of the foundation successes of the Gundam franchise). While Saegusa produced only a little more anime music, his Zeta Gundam soundtrack is still considered a classic among otaku. For many of them, Saegusa and Hisaishi were the first to inspire the idea that anime music could be of very high quality.
Meanwhile, Kenji Kawai was producing scores for series such as Blue Seed, Patlabor, and Ranma 1/2. While few of these scores were groundbreaking, they were almost all solid works of music. Kawai was arguably the first composer to produce a number of anime soundtracks and achieve at least a modicum of popularity within the otaku community while doing so.
During the 1990s, Yoko Kanno garnered some interest with her soundtracks for Escaflowne and Macross Plus, but it was her soundtrack for Cowboy Bebop in 1998 that made her extremely popular among anime fans. Kanno is by far the most popular composer in the anime field today.
Meanwhile, Taku Iwasaki (the Rurouni Kenshin OVAs, Witch Hunter Robin, Read or Die TV) and Yuki Kajiura (Noir, .hack//SIGN) have both produced several well-respected soundtracks in the late 1990s and 2000s.