Can astronomers directly detect black holes? Are there any methods of direct detection for black holes?
I'm not referring to gravitational lensing, or measuring the orbits of a star in a binary pair.
Is there any way of directly 'seeing' them? 
 A: A colleague in astronomy had a student a few years ago who did a calculation about the possibility of primordial black holes, created in the Big Bang. If the size of these was just right, they could be evaporating into nothing due to Hawking radiation right "now" (scare quotes because this would necessarily include distant black holes that evaporated many years ago, whose light is just reaching us now). The last burst of Hawking radiation for these would look basically like a faint gamma-ray burst, in which case it ought to be directly detectable.
I'm not sure of the current status of this-- their preliminary result was, if I remember correctly, that you might be able to test this by measuring the probability distribution for gamma-ray bursts of the appropriate size and duration, but we didn't have any telescopes capable of picking them up at the time. I'm not sure if that's changed or not.
Anyway, that would give you a direct way to detect black holes of a certain size, though they wouldn't be around after the detection, so it might not really fit the spirit of the question...
A: As far as anyone knows, with the exception of the Hawking radiation that Chad mentioned, black holes don't emit anything that we could use to detect them directly. And Hawking radiation from most black holes is so weak that it'd be impossible to detect from any significant distance, except for the final burst of radiation when it finally evaporates.
So the straightforward answer would be: generally no, black holes can't be detected directly.
However, any black hole that is close enough to another star will have an accretion disk, and that can certainly be detected because it gives off a lot of X-rays and gamma rays.
A: Are you aware of the Cygnus X-1 x-ray source? This might give you an example of "seeing" a black hole. This is an example of an accretion disk x-ray source, discussed in other answers here.
A: There is some work on the subject by professor Narayan. As David said, a black hole will usually have an accretion disk that radiates a lot of energy. That energy comes from the gravitational potential energy. As the material of the disk falls towards the central object, it transforms its potential energy to radiation while the material is on the disk and interacts with the rest of the disk. On the final part of the infall of the material to the central object, where the infalling material gets separated from the interior edge of the disk, the gravitational potential energy is transformed to radiation only if there is a central object with a surface. If the central object is a black hole, then there should be a lot of energy missing from the center of the accretion disk.
This is again, not a direct detection, in the sense that you don't see the black hole itself. The only thing that a black hole can emit is Hawking radiation, but that radiation is relevant only for non-astrophysical black holes (collapsed stars or centers of galaxies), like the primordial black holes that Chad mentioned (which are small) or mini black holes that could be created in an accelerator or from cosmic radiation.
A: If you were close enough, or the black hole big enough, beams of light passing near it
would distort the image of objects more distant than the BH, also some of this light would 
disappear down the BH's event horizon. So yes, if you had a big enough telescope you could see
it. But the size of BHs relative the their probable distance means this telescope won't be
around for many many years. In a similar vein, wouldn't looking at a BH against the cosmic
background radiation do the same trick? You'd see Hawking radiation at a much lower
temperature than the CMB (unless the BH is very small). But, again given the actual 
size/distance of BHs, the needed resolution is far away from being achievable.
A: If a black hole is illuminated along our line of site by a source behind us, it might be possible to see its shadow against a bright background.  
A: There are two levels at which this is being asked I think.  The animations in the link below illustrates what black holes appear as in front of stars or the galaxy.  It also portrays the black surface which demarks this as a black hole.  Will we ever see a black in this form?  Maybe, or we might get some better signature of this.  Much higher resolution could demonstrate angular dependencies with accretion disks around black holes.  Similarly it is possible that the super massive black hole in this galaxy could be resolved in a similar way, at least to measure optical lensing of distant stars.
http://jila.colorado.edu/~ajsh/insidebh/
There is of course the prospect that black holes exhibit some quantum properties around the 1-10 TeV range of energy.  This is the large extra-dimension prospect, where maybe Calabi-Yau manifolds “unfold” to larger scales at higher energy.  In doing so physics at the TeV range of energy may have some small amplitudes corresponding to graviton and black hole physics.  At low energy the physics is QFT, while at higher energy it assumes some gravitational aspects.  We might think of this as doing QFT on the surface of the AdS, but where at sufficiently high energy we get some small sampling of gravity in the interior of the AdS.  
The extent to which we probe the interior of the AdS might be modeled with a simple field.  We might think of this as the manifestation of an auxiliary field which converts one sort of physics at low energy into another at high energy.  The transverse modes of a string on the boundary define the CFT there, and these assume a certain value on this boundary.  In the interior the longitudinal modes are nonvanishing, but vanish on the boundary.  The noncommutative coordinate uncertainty principle $\Delta X^+\Delta X^-~\simeq~{\ell}_s$ $=~4\pi\sqrt{\alpha’}$ is then off quadrature near the boundary of the AdS.  This auxiliary force is then analogous to the squeezing operator in quantum optics. 
As a toy model consider a $n+1$dimensional “spacetime plus R” space. This additional dimension is a space with a gauge connection or potential $A~=~\phi$ which defines a force $F~=~-d\phi/dx_5$, for $x_5$ a parameter on this fifth dimension.  An obvious simple case for a field in a one dimensional chord of length $L$ would be where $\phi~=~\pi x_5/L$ as a model similar to a constant gravity on Earth.  This “force” then adjusts the noncommutative uncertainty principle, which is in a complete squeezed state on the AdS boundary.  We then at high energy adjust this slightly by pushing the conformally flat spacetime on the boundary into an additional dimension in the interior of the AdS.  We might then gets some signatures or small amplitudes of quantum gravity, or quantum black hole physics, in LHC scale experiments.
The prospect is not entirely absurd, for at the EW symmetry recovery level QFT may exhibit a renormalization group flow to high energy.  This flow is then logarithmic in energy, and so the difference in scale from $10 TeV$ to $10^{16}TeV$ is not that dramatic.  It then might be the case that small amplitudes for quantum black hole physics could turn up in the LHC.
A: And, in 2018, we might be able to eventually actually see them:  https://eventhorizontelescope.org/
