Update 7/26/12 :
font-smooth:always was a Mozilla-only property that has been depreciated. I've started maintaining a Coderwall entry about web typography tips that might alleviate some of these pain points.
I've been dealing with this issue a lot lately because I'm incredibly picky with colors and type, and there are a variety of things you can try. They have different effects cross-browser and cross-OS, so I highly recommend conditioning these to only apply to IEs and Windows machines once you've fleshed out how it renders in OS X. Here are some techniques I've gathered:
CSS2+: Some browsers look good if you just apply a text-shadow. What settings to use varies, with some people liking something like
text-shadow: rgba(0, 0, 0, 0.0078125) 0px 0px 1px; (as suggested in this StackOverflow) and others a simple
text-shadow: #fff 0 0 1px; (this only really works well in Chrome). There is also this technique (
text-shadow: 0 0 1px rgba(0,0,0,0.3); where the RGB matches your
color), which is incredibly subtle but enough of a difference that it worths.
I've had both success and failure with these methods across different typefaces and implementations (from Google-hosted fonts to your default sans-serifs), so test every tweak in every browser for each typeface and size you're using it at.
Something else I had an issue with (due partly to my OCD) is the strain this put on my eyes and the bias I have towards thin text. Have friends or colleagues look at it during different stages -- not too often but more than once or twice. Keep
/*comments*/ in your CSS so that you know what worked best where and what each of your testers thought along the way.
Above all else: work with subtleties! Don't feel like tweaking that .02 to .022 isn't going to make a noticeable difference because for this, it probably will.
Your users will love you for it.
Edit: Berin points out issues you might have with letter-spacing after testing some of these out, so use Lettering.js in conjunction with Kern.js to kern and otherwise make minor tweaks to headlines.