What if your randomly formed groups are clearly not similar? What if, before you begin the data collection for an experiment, you randomly divide your subject pool into two (or more) groups.  Before implementing the experimental manipulation you notice the groups are clearly different on one or more variables of potential import.  For example, the two (or more) groups have different proportions of subjects by gender or age or educational level, or job experience, etc.  What is a reasonable course of action in such a situation?  What are the dangers of discarding the original random division of the subject pool and dividing the pool again?  For example, are the inferential statistics that you might calculate based on the second set of groups in any way inappropriate due to the discarded first set of groups? For example, if we subscribe to discarding the first division of the subject pool into groups, are we changing the sampling distribution that our statistical test is based on? If so, are we making it easier or harder to find statistical significance? Are the possible dangers involved in repeating the division of subjects greater than the obvious danger of confounding due to group differences in educational level, say?
To make this question more concrete, assume for the sake of this discussion that the topic of the research is teaching method (and we have two teaching methods) and the difference noted between the two groups of subjects is level of formal education, with one group containing proportionally more people with highest educational attainment of high school level or less and the other group containing more people with some college or a college degree.  Assume that we are training military recruits in a job that does not exist in the civilian world, so everyone entering that specialty has to learn the job from scratch.  Assume, further, that the between group imbalance in previous educational attainment is statistically significant.
Parenthetically note that this question is similar to What if your random sample is clearly not representative?.  In a comment there, @stask perceptively noticed that I am a researcher not a surveyor and commented that I might have gotten more relevant answers had I tagged my question differently, including "experiment design" rather than "sampling." (It seems the sampling tag attracts people working with surveys rather than experiments). So the above is basically the same question, in an experimental context.
 A: If you just do a new randomization of similar type to the previous one (and allow yourself to keep randomizing until you like the balance) then it can be argued that the randomization is not really random.
However, if you are concerned about the lack of balance in the 1st randomization, then you probably should not be doing a completely randomized design in the first place.  A randomized block or matched pairs design would make more sense.  1st divide the subjects into similar groups based on the things that you are most concerned about, prior education in your example, then do a randomization within each group/block.  You will need to use a different analysis technique (randomized block, or mixed effects models instead of one-way anova or t-tests).  If you cannot block on everything of interest then you should use the other techniques of adjusting for covariates that you do not block on.
A: I do not think sampling should be adjusted because of chance imbalances.  Adjusting creates complications that can be worse than any problem you think you might solve.  If in the end you have covariate imbalances there are ways to adjust for them.  See this book by Vance Berger for example.
Selection Bias and Covariate Imbalances In Randomized Clinical Trials
A: In my opinion, drawing and redrawing a sample in response to not looking "random enough" doesn't generate any bias so long as 


*

*You aren't making that decision based upon your outcome of interest and 

*You condition on your observables.


We know that bias is introduced by selection on unobservables. As long as you perform selection based upon observables and those observables are conditioned on in the analysis, you're okay.
Is this still random? I say yes, it is a random sample, conditional on the observables. This is what matters. We need to be careful about how we are defining "random" and when we are precise, we see that randomization is still here.
How might you condition on them? Linear models are a standard method. Matching is a nice non-parametric procedure, but requires a bit more thought that you might imagine (I'm biased to be fond of genetic matching algorithms).
I would note that the conditioning process smooths out or takes into account (depending upon how you want to look at it) the imbalances present. So you don't need to redraw your sample. The only time that you might need to do this is if you want to do matching and you don't have common support between the two groups (you have some college graduates in one group and none in the other, for example).
Blocking or stratifying is really about variance reduction (unless your interested in non-homogeneous responses along some dimension), not about bias.
Since you still have randomization, no adjustments need to be made in your testing procedures. (Though, if you're doing matching, be sure to use variances estimators designed for matching; see Abadie and Imbens (2006))
A: An experiment requires a control, but it doesn't need to be structured as "end result of one randomly assigned group" vs. "end result of another randomly assigned group". You are concerned that the composition of the groups is unbalanced on one particular attribute that you believe has an outsized influence on the end result; perhaps a different structure would make this irrelevant. The experiment may be set up as a before-and-after for each teaching method group, or it can be a regression where one independent variable is the teaching method. If you have really strong reasons to believe that this one particular attribute outweighs all others, you could even take your population of participants and separate them into groups according to education level, and then within each group randomly assign participants to the control and test groups.
A risk is that while you've picked out one key driver of different outcomes, there may be others lurking in there you haven't detected, and which you might be subverting. 
