Interface defining a constructor signature?

You can't.

Interfaces define contracts that other objects implement and therefore have no state that needs to be initialized.

If you have some state that needs to be initialized, you should consider using an abstract base class instead.

A very late contribution demonstrating another problem with interfaced constructors. (I choose this question because it has the clearest articulation of the problem). Suppose we could have:

interface IPerson
{
    IPerson(string name);
}

interface ICustomer
{
    ICustomer(DateTime registrationDate);
}

class Person : IPerson, ICustomer
{
    Person(string name) { }
    Person(DateTime registrationDate) { }
}

Where by convention the implementation of the "interface constructor" is replaced by the type name.

Now make an instance:

ICustomer a = new Person("Ernie");

Would we say that the contract ICustomer is obeyed?

And what about this:

interface ICustomer
{
    ICustomer(string address);
}

You can't. It's occasionally a pain, but you wouldn't be able to call it using normal techniques anyway.

In a blog post I've suggested static interfaces which would only be usable in generic type constraints - but could be really handy, IMO.

One point about if you could define a constructor within an interface, you'd have trouble deriving classes:

public class Foo : IParameterlessConstructor
{
    public Foo() // As per the interface
    {
    }
}

public class Bar : Foo
{
    // Yikes! We now don't have a parameterless constructor...
    public Bar(int x)
    {
    }
}

As already well noted, you can't have constructors on an Interface. But since this is such a highly ranked result in Google some 7 years later, I thought I would chip in here - specifically to show how you could use an abstract base class in tandem with your existing Interface and maybe cut down on the amount of refactoring needed in the future for similar situations. This concept has already been hinted at in some of the comments but I thought it would be worth showing how to actually do it.

So you have your main interface that looks like this so far:

public interface IDrawable
{
    void Update();
    void Draw();
}

Now create an abstract class with the constructor you want to enforce. Actually, since it's now available since the time you wrote your original question, we can get a little fancy here and use generics in this situation so that we can adapt this to other interfaces that might need the same functionality but have different constructor requirements:

public abstract class MustInitialize<T>
{
    public MustInitialize(T parameters)
    {

    }
}

Now you'll need to create a new class that inherits from both the IDrawable interface and the MustInitialize abstract class:

public class Drawable : MustInitialize<GraphicsDeviceManager>, IDrawable
{
    GraphicsDeviceManager _graphicsDeviceManager;

    public Drawable(GraphicsDeviceManager graphicsDeviceManager)
        : base (graphicsDeviceManager)
    {
        _graphicsDeviceManager = graphicsDeviceManager;
    }

    public void Update()
    {
        //use _graphicsDeviceManager here to do whatever
    }

    public void Draw()
    {
        //use _graphicsDeviceManager here to do whatever
    }
}

Then just create an instance of Drawable and you're good to go:

IDrawable drawableService = new Drawable(myGraphicsDeviceManager);

The cool thing here is that the new Drawable class we created still behaves just like what we would expect from an IDrawable.

If you need to pass more than one parameter to the MustInitialize constructor, you can create a class that defines properties for all of the fields you'll need to pass in.