Q:

Advice on using OCR on an image of a blackboard

I'm trying to get an image of a blackboard readable by OCR.  Naturally, most OCR software doesn't like dirty images.  What image processing should I try to put the image through to clean the image up? 

A:

Have you tried the OCR software yet?  It's likely that the OCR software is well suited to reading what's essentially already a black and white image. 
However, if you were required to do so you could try to:

Threshold the image.  

Essentially take a greyscale version of the image and turn it into black / white pixels

Perform Binary Dilation to grow the remaining objects
Perform Binary Erosion

The idea is by dilating then eroding you would remove any rough / noisy edges and then you can pass the skeletonized image to the OCR. 
There are probably plenty of methods to achieve a similar result.  Given that there are entire books devoted to computer vision this answer will hardly do them justice. 
The only texts I have are from 1997, but surely there's been more written on the subject since. 

Algorithms for Image Processing and Computer Vision - J.R. Parker
Digital Image Processing - Gonzalez / Woods 

