This question has actually been an issue over the course of Japanese history and there have been several advocates for reform since the Edo period.
You can have a good read about it here:
Japanese script reform - Wikipedia, the free encyclopedia
At the start of 1946 there were future plans to completely abolish kanji, but with the advent of computers and IME it has helped preserve kanji.
Even so, there are some organizations in Japan that promote the use of rōmaji, romanization of the Japanese alphabet or the use of a kana-only script:
Nippon-no-Rômazi-Sya (An organization promoting the use of rōmaji) (Japanese)
The Society for the Romanization of the Japanese Alphabet (Japanese)
Kanamojikai (An organization promoting the use of a kana-only script) (Japanese)