Lots of Japanese are aware that the Yasukuni Shrine visits by Koizumi do cause seriously strained relationships with Japan's continental neighbors, especially China (where violent protestors made Koreans look like the sane ones). It's actually not too hard to find people opposed to the visits.
But the right-wingers in Japan continue to visit, claiming that to not do so would be an insult to the war dead. So it is interesting to see in today's Kyodo that Nippon Izokukai (the Japan War-Beareaved Association), headed by former LDP Secretary General Makoto Koga and a major supporter of that party, has issued a statement on the PM visits.
[Having prime ministers paying homage at the shrine] has been an ardent wish of the association and we appreciate it very much but, at the same time it is most important that the spirits of the war dead rest in peace. It is necessary to give consideration to neighboring countries and obtain their understanding.One major policy of the association is to have regular Yasukuni visits by Japanese prime ministers, but the statement urged Koizumi, in an unusual move, to consider the criticisms of neighboring countries such as China and South Korea.
Kyodo reports that the statement also said politics should not be brought into the argument about whether Class-A war criminals should be separately enshrined from the war dead.
The association, however, expressed their opposition to the building of a separate shrine, an act which some have said would resolve the issue. The association considers Yasukuni Shrine as "the only memorial facility for the spirits of the war dead" and thus opposes establishment of any new facilities for the war dead in the future. They also oppose separate enshrinement of the Class-A war criminals.
Still, it's important to hear that even many on the right in Japan recognize that it's not in Japan's best interest to go pissing off the neighbors.