I have nothing to offer.
(There's another egg from Lucy as I write at 9:20 p.m. Excuse me, dear...)
The gifting I have done for couples has been through showers given by outside parties. Because I have some connection to everyone who invites me to one of these things, I'll call a mother or someone else in the know to ask what the couple wants or needs. If one of those people were to tell me that the couple would like cash, I wouldn't be at all offended. I appreciate the flexibility of cash myself.
However, I've never heard of a bride asking for her own gifts, and I've never encountered a direct request for cash.
UPDATE: As brides and grooms become older and better established before they marry, we might need to find a way for them to make that request without embarrassing themselves. The last four gifts I've given have been gift certificates -- one to Lowe's, two to Amazon, and one to the shop of registry. A check would have been more convenient for me and more welcome to a couple who would, for instance, like to purchase things on E-Bay.
Kate, you married recently. What's your opinion?
Dr. James Joyner was married fairly recently. What does he think?
UPDATE: Dr. Joyner answers:
I don't think we said anything about gifts on the invitation itself, since those things tend to be formal. On our wedding blog and eVites, though, we put a note to the effect "we've got enough crap, don't buy us anything" but established a couple wedding registries at Macy's, Bed Bath and Beyond, and Home Depot.UPDATE: Kate's answer here.