I enjoy any appearance of Godzilla (except the old cartoons, sorry) but like the most recent cycle the most, probably because I haven't 'wore them out' yet. The 1999 to 2004 cycle did their best to reinvent Godzilla movies and inject new talent and story ideas into the series.
Also, this may be a merciless plug for my blog, but I really think it could be useful here - a guide to all the Godzilla films currently out on DVD in the US, UK and Hong Kong. Of course, they are all out in Japan, but with no english translation, subtitles or audio tracks.
I keep this entry updated with every new release.
My Godzilla DVD checklist