Sometimes the "instant" is an essential part of the request. For example, I want all requests timestamped. That's easy to do now, because I can register a single controller decorator (like a "before filter" in Rails) that checks now() for every request and adds it to the request. Either it passes this as a separate parameter (like in my example code here) or it merely adds this to the Request object. (You can decide to store this as a request attribute or have a new Request wrapper that includes some extra, common information.) Now it is the same as adding headers to a request. No pollution.
Your domain layer never cares about the concept of "now" or "recent", but only of "relative to some instant" or "relative to some date". This simplifies the domain and makes its behavior more referentially transparent. Easier to reason about; easier to test; still easy to use.
Of course, you will have some domain concepts that don't need to know about time at all, and their clients simply don't pass the time along. No problem there.
Any place that the domain wants a "clock", try replacing that with either a single instant or a stream of instants. I think you'll find the code to be equally clear and the tests to be less complicated. The Clock is simply a complicated way to get a single instant. :) You'll notice the difference when you write the tests.