I always thought Ritsurin Koen on Shikoku was one of the big 3, but maybe I just wanted it to be since I've actually been there. Anyway, it, too, is too far for a quick visit.

I'll second Katsura Rikkyu (referred to in a guide I have as Katsura Imperial Villa) as worth a visit in Kyoto. You have to apply to go visit (to the same people you would if you wanted to see the inside of the Imperial Palace), but as a foreigner it's much easier for you to visit than for a Japanese national. Shugakuin, another Imperial villa in the northeast part of the city, was lovely in autumn, but I'm not sure about spring. Any detailed guidebook should provide you with some explanation of how to apply for these places.

Despite living for a year in Kyoto in college, I never visited Saihoji (Kokedera, the moss temple) as the whole application/seated prayer/writing out sutras thing seemed a bit much, particularly at that age. Has anyone been? Is it worthwhile?