Cohen's d in non-significant results I did several independend t-tests. Some of them are significant, some of them are not. 
Next, I computed Cohens $d$ for effect size. 
For significant results, Cohens $d$ feels intuitively reasonable (How "large" is the found effect between both samples?). But how do I properly interpret Cohens $d$ in non-significant results?
 A: Cohen's d can help to explain non-significant results: if your study has a small sample size, the chances of finding a statistically significant difference between the groups is unlikely, unless the effect size is large.
It's probably a good idea to include a confidence interval for your Cohen's d since the effect size based on your sample is still an estimate of the 'true' effect size. 
This can be done easily in R using the tes() function in the compute.es package:
library(compute.es) 
tes(t=??, n.1=??, n.2=??)

Here is a useful short article on effect size thresholds: thresholds for interpreting effect sizes2.
And here is one on combining effect sizes with significance test interpretations (see especially sections 4 and 5): It's the Effect Size, Stupid: What effect size is and why it is important.
Here is a related post on how to interpret the confidence interval of Cohen's d in case you choose to find that: 
Why is the p-value for Cohen's $d$ not equal to the p-value of a t-test?.
A: Adding to @Jordan's answer above, in my practical experience in a federal agency that funds a lot of program evaluations, if we see results that are not statistically significant at conventional levels, we don't typically bother to even read anything into the effect sizes. The idea being here, if the effect is not statistically significant, we cannot rule out random chance that there is an effect at all, so however large the effect size is, it may not be different from zero upon repeated sampling. 
There are, clearly, problems with "statistical significance", especially when we are talking about small sample sizes. I don't really want to get into philosophical arguments here, and I'm certainly not one to defend an approach based on p-values. This is to just give you some practical context for how (some) researchers and evaluators in the federal government respond to this sort of thing.
