Turns out that there are quite a few; I had time to try out the most popular ones: mint.com, wesabe.com, and geezeo.com.
Mint.com is by far the most polished. The site looks nice, with rounded transitions and flash animations. While in most cases, the interface simply makes the experience a bit more pleasant, their spending trends visualizer is great, letting you drill down into a pie chart of spending categories and see the corresponding transactions. (It also feels more responsive than Quicken on my computer, but that's saying more about Quicken than about Mint.com.) Wesabe.com seems to have pretty good visualizations, too, though it feels less polished and responsive overall. Geezeo is the least pleasing to the eye, and I actually could not find any visualization tools.
The coolest feature of the online sites is that they can automatically download the financial data from your accounts. It's been my job to regularly download transactions from all of our banks and put them into Quicken. With the sites, they will actually do this automatically and will always have up-to-date information. So with the Quicken method, it was pretty infeasible to check our budget progress more than once a month (in fact, I'd often go many months without the downloads, even when I wasn't in the hospital), with the online sites, you can really check your budget progress weekly or even daily.
The process of linking accounts is slightly different for each site. Mint.com has a large list of financial sites it supports and you simply enter your login credentials. They don't have a login for our home bank; I'm starting to suspect that this is because it uses a (world's dumbest) CAPTCHA at login time. But they seem to support every other bank in our area, as well as every other financial institution (loans, IRAs, mortgages) we use. Wesabe.com can download transactions from a limited number of sites; for everything else, you can get a Firefox plugin that can record your login and download interactions and upload data to Wesabe. Clever, though more cumbersome than simply supplying your credentials (and it still won't work with our home bank). Geezeo's solution for unsupported banks is simply to let you upload an .ofx file manually, which might be the best approach.
Once the transactions are imported, the sites let you categorize them, with some automation built in. Mint.com has the best automation right off the bat, classifying probably 80-90% of our transactions automatically. Wesabe requires that you enter your own tags, but will apply them to similar transactions in the future. Geezeo follows a similar system.
The tag structure in Wesabe and Geezeo is free-form: you enter as many tags as you want for a transaction, and Wesabe will even display all your tags in the familiar tag cloud format. Mint.com, on the other hand, has a more rigid structure, with some user extensions built in (reading online reviews, it looks like the extensions are a recent addition). It has a fairly detailed two-level category hierarchy. You can add your own categories, and it even comes with a large bank of suggested ones, but you can't delete the built-in ones. It also allows you to assign tags to transactions; tags can span multiple categories and a transaction can have multiple tags. So in the end, you can get a similar structure to Wesabe and Geezeo, but there's a separation of primary and secondary properties of a transaction.
You can then use these tags and categories in visualization tools, or to set up a budget of spending targets. Here, all three sites leave a lot to be desired. Mint.com allows to create a spending target for a category or a sub-category. Wesabe lets you use a tag, and Geezeo lets you combine two tags (using an OR). So, for example, Mint.com allows you to tag things as Reimbursable (it's even part of a default tag set), but you can't create a budget for non-reimbursable travel expenses. (Though you can query them in the visualizer.) And in Wesabe and Geezeo, I was hoping to create separate budgets for "clothing nikita" and "clothing lenore," but neither will let me do that. (I'd have to basically tag things as "clothing clothingnikita" and "clothing clothinglenore.") My hope is that, given that all of these sites are fairly recent, at some point at least one of them will get around to implementing some decent search operators.
We finally settled on Mint.com as our main site. It has a nice interface, it painlessly supports all of our accounts except for the checking one (we actually looked into switching banks and found a better deal at another bank in town that is "minty"), and it has an iPhone interface which looks pretty slick. It seems like it would be pretty useful for day-to-day planning. One nice thing about the budget display in Mint.com is that it not only shows you how much of the budgeted amount you have spent so far, but also a line that shows where you should be given how far into the month/year/etc. you are, so we can see, for example, that our food expenses for this month have actually been lower than expected.
I'll have to see how well it will work for a monthly evaluation of spending, but I'm pretty sure it will save me time over using Quicken. And it's pretty cool to see all of our financial accounts added up in one place to calculate our "Net Worth" (which is heavily negative, since it includes our mortgage).