Yakuza 3 is a fun game, although it's probably a less fulfilling experience than the second entry in the series - which, being released on PS2 after the PS3 and the other next generation consoles began to dominate the market, was mostly overlooked. This third entry into the series adds a few new minigames (although as noted by other reviewers some of these have been cut from the European/American version of the game) and has had a visual overhaul.
I'm about ten hours into the game and I've only scratched the surface of things to do. There are some frustrating elements: cut-scenes are sometimes interminable, and the missions involving Kazuma sorting out the problems of the orphans in his orphanage are sometimes more tedious than fun. However, there is plenty to explore and the combat is often most satisfying - even though it's usually a case of using no more than three or four buttons (block, strike, strong strike, grab/throw). Early in the game, combat is limited; but it gradually opens up as you progress through the narrative, with new moves and weapons being introduced.
The game's narrative is strong; in fact, the story - focusing on issues of community and economic regeneration - has more resonance than most of the Hollywood films I've seen in the past year or two. Kazuma has always been an interesting, conflicted protagonist, with his abandonment of the yakuza lifestyle comparable to Michael Corleone's relationship with the cosa nostra in The Godfather, Part III ('Just when I think I'm out, they pull me back in').
I bought the game knowing that some of the content was cut (the hostess bars, some quiz mini-games, etc) and I'm thoroughly happy with the game. However, I wish Sega hadn't cut the game; and their excuses for doing so seem to do nothing but dig them a bigger hole with fans. Nevertheless, I think it's important to support the series, otherwise we may never see an English version of Yakuza 4 - with cuts or not. The Yakuza games sell poorly over here as it is.
In sum, Yakuza 3 is a very good game, even if it is not equal to Yakuza 2. Hopefully, Yakuza 4 will be released over here, as advance word on the game (from Japan) seems to be quite positive. If the idea of cut content bothers you, I'd recommend renting the game before buying it; however, in my playthrough (so far), I have to say that although I'm aware of some content being cut I haven't been stuck for things to do in the game, and I can't say that my enjoyment of the game has been impaired.
Most of all, playing Yakuza 3 has made me dig out some of my old Kinji Fukasaku Yakuza movies. That can't be a bad thing...