I think "acceptance" is actually a nuanced problem that is fuzzy, social, and negotiable. Using tests to mediate this problem is a bad idea, in my opinion. I'd rather see "acceptance" be done through face-to-face conversations before, after, and during development of code, centering around whiteboard sketches (earlier) and manual demonstrations (later) rather than automated tests.
That said, we still need to test the behavior of the software. But this is a programmer concern, and can be done with programmer tests. Tools like Cucumber shift the burden of "the software being right" to the customer, which I feel is a mistake. It's our job as programmers to (a) work with the customer, on the customer's terms, so we build the right thing, and (b) make sure it actually works, and keeps working in the future. TDD helps us do it, and it's our responsibility, not the customer's.
I don't know if this is very clear. To rephrase: "acceptance" should be a conversation, and it's one that we should allow to grow and change as the customer sees the software and refines her understanding of what she wants. Testing is too limited, and too rigid. Asking customers to read and write acceptance tests is a poor use of their time, skill, and inclinations.
I have more here: