Well, my husband was born in this country, but in the eastern US city from which he comes, there is a very strong tradition of giving money at the reception, and this can be in addition to in-kind gifts brought to the reception, sent ahead of the wedding, sent after the wedding, or brought to the shower. The parents of the bride seem to arrange some sort of large decorated basket for gifts to quietly, anonymously be placed in. There is a name for that basket, but I don't remember it. Then the bride also carries a purse and takes money in exchange for dances.

is how I reacted internally the first time I saw that money dance thing. Where I grew upon in the south, receptions were very tame and reserved. Not until my adult life did I ever go to a Southern wedding reception with alcohol, dancing, or money being flashed.
Well, wedding customs are indeed changing, sometimes not for the better in my opinion. I do think there should be an option for relatives or friends of the bride and groom to spread the word that they really could use money. In this day, it can be pretty hard to even pick a wedding present for others who are combining two households of belongings and probably need to weed out duplicates---not get more of them. As a guest, it would make me happy to give money, especially as I do not consider myself to be great at picking out gifts for others. (Sad about that, though). However, as matters stand, I think that for every ten people you directly tell that you would prefer cash, you are bound to grate the nerves of at least one, probably two. When else in life is it acceptable to ask others to give you money? Maybe when you are four years old and would like a gum ball from the machine, but not much beyond that.
InDebtinDC & Aleta, I have some friends who are an Asian couple who have been in the US for less than 10 years. I am forever being a cultural consultant to them, especially to the woman who earnestly wants to do everything just right. It is hard to explain that there is not one clear cut answer to every situation. There are many acceptable ways to act, things to say, ways to dress, salutations to give, presents to give, meals to serve, etc---in so many situations. I don't see, Aleta, how you could say what is "the" American tradition in many cases. There is a compendium of traditions, and those are going to be ever changing. Actually this is true in much of the world now, I suspect. The US is far from being the only country with large numbers of immigrants and there are other forces that change customs besides immigration.