For those not doing a continual volume of link-building or link research, the loss of a free tool like Yahoo's can hurt.
At PubCon, Dixon showed me a new, more user-friendly interface for Majestic that's a little like SiteExplorer on steroids... extremely nice, but unfortunately not free. I don't know whether that interface will be made available for analyzing your own site (or client sites you have access to).
Bing's Duane Forrester (bingdude) mentioned in his PubCon presention that he hopes he can persuade Bing to come out with a tool like SiteExplorer that won't be limited to just data about one's own site. It would be nice to have such a public tool available.
I'm not a fan of SEOmoz's Open Site Explorer, so I can't say much about it. Pretty limited capabilities, as I remember, with data that's about a month behind. I think SEOmoz has the most to gain by providing a free tool, so it's possible it may get better. Anyone experienced with it? I don't think there's anything else, unless you count Blekko, which is basically self-censored and therefore limited, particularly for competitive research where you want to see why competitors are ranking.
It would be ironic if Google were actually showing us most of the links it counts.