They would usually tell me to use my imagination. They think that once they give you the freedom to do what you want, you'll actually deliver the best logo they've ever seen.
That's usually not the case.
I'm noticing that once I have some limitations and strict requirements, I approach the project like a maths problem. And since I've always loved maths, I love the analogy as well. Here's my reasoning.
In a maths problem like this:
x + y + z = w
the designer will always have one or two of the parameters, so the task will look like this:
1 + 2 + z = w
With this, somewhat more clear, problem at hand, we need to find w. In order to find w, we first need to find z. Yes, I know that's quite a tough maths problem, but that's roughly how I approach logo design projects. The logo is w. But there are other elements I'm missing - maybe the client hasn't defined something in his brief. I may have to deduct some information on the basis of his request/brief. In a way, I have to reverse-engineer his vision about the logo and process that through the brief.
That's where things get tough. I don't want to create just any logo. I want to create an incredibly smart and clever logo. This means, I add another restriction to my problem:
1 + 2 + z + m = w
So, with my clients' expectations and my own limitations/restrictions the task seems pretty clear - there's only ONE logo that fits all these requirements, and I have to find it! If I don't find it in a certain amount of time, that doesn't mean this logo isn't possible to create. The world is a limitless thing, and virtually everything potentially exists! Ergo - my perfect logo exists too! I will find it! I have a detective spirit that doesn't give up when it meets some tough problems.
So, in my case, I don't want my clients to give me freedom. I want them to have specific requirements and I want them to know what they want, but not limited to abstractions only!
Of course, I understand the fact that non-designing clients don't know exactly what they want. They have the vision, the urgency inside, the business need, and indeed their only means of expressing that is through abstract words. I'm guessing that makes the designer's task even tougher. Usually clients with such an approach may request lots of revisions, until they 'click' with the current logo version.
It's up to the designer to lead the client by asking questions. Sometimes the best answer is held by the client. If the client is willing and patient enough, s/he will stay through a questions-and-answers process until the designer has a much clearer idea about the desired logo. This will also help the client define his/her needs. Oftentimes, abstract descriptions don't mean that the client is unsure about the precise logo, instead, they just can't formulate it. Your questions can lead them to a much more specific vision/potential design. This may include typefaces, color schemes, graphic elements, etc.
So, help your clients and do the problem!