Dichotomizing continuous variables at their optimal cut-off for clinical interpretation In medical context, when presenting results from a binary outcome with a continuous predictor, the OR (odds ratio) can be difficult to interpret.
Example:
A doctor does a study in which he wants to see if high blood pressure (continuous) can potentially increases the risk of heart attacks (binary heart attack no/yes).
He does so by retrospectively looking through patient records and noting whether or not they had a heart attack and what their blood pressure was.
He performs a logistic regression with the continuous blood pressure variable and gets an OR of 1.01.
Now the question is. This OR does not seem very drastic and may be difficult to understand for some clinicians. The doctor therefore does a ROC analysis to see at what value the sensitivity and specificity of blood pressure is highest to predict a heart attack. He notices this is at 150 mmHg (ignore the context, let's assume this is the best value for his purposes).
He regresses again, with heart attacks and the new dichotomized BP above or below 150 mmHg and gets an OR of 5.
This is a lot easier for clinicians to understand. If your blood pressure is above 150 mmHg, your odds of having a heart attack increase 5fold (this is of course not to say your probability increases 5fold).
My question is, would this be a correct way to handle this data? Knowing statistics, there are almost always pitfalls to even the slightest adjustments so I want to hear your input.
I should say I understand the problems of dichotomizing continuous data, e.g. it's pretending that the difference changes from no risk to massive risk from 1 mmHg to the next, but if he was to relay some information to his peers of his findings in a digestible manner, would this be possible?
 A: For digestibility, use examples from representative situations: in your example, maybe comparing risks at BP of 160 versus BP of 120.
For an approach that can take into account the multiple predictors typically important in clinical studies, use a nomogram. It provides a graphical tool to show how predictor values affect outcomes. The rms package in R provides tools for constructing nomograms from fitted regression models.
This particular approach that you propose:

The doctor therefore does a ROC analysis to see at what value the sensitivity and specificity of blood pressure is highest to predict a heart attack. He notices this is at 150 mmHg... He regresses again, with heart attacks and the new dichotomized BP above or below 150 mmHg and gets an OR of 5

is ill-advised for reasons besides the general issues of dichotomization that you acknowledge. For one, use of sensitivity and specificity tends to involve a hidden assumption that false-positive and false-negative classifications have the same costs. For another, once you use the data to set the cutoff, the assumptions underlying p-value and confidence-interval calculations no longer hold.
A: Another issue is that the relationship between the IV and the DV (here, BP and heart attack risk) might not be linear. I would think this sort of nonlinearity would be quite common in medical fields. Indeed, this is sometimes given as a reason for categorizing the continuous variable (albeit into more than two categories). But this isn't good. A better method is to use a spline of the IV. 
A: Dichotomizing a continuous covariate is ill-advised, as has been noted by other users.
One strategy I employ is to rescale the predictor to something more reasonable.  1 mmHg may not be a very meaningful scale on which to interpret changes in BP.  But, if you rescale the predictor so that a difference of 1 unit represents say a difference 10 mmHg then things become a little easier to digest and the odds ratio will be more appreciable and have the following interpretation

For every 10 mmHg increase in blood pressure, the odds of MI increases by a factor of $\exp(\beta)$.

A: Similar to EdM's answer, a marginal effects plot is a useful way to showcase the relationship between a clinical measurement and outcome while holding other variables constant. These plots are helpful because they show the relationship between the predictor and outcome, so if the outcome is nonlinear, physicians can easily see this and interpret it appropriately. Here is an example from Frank Harrell's book Regression Modeling Strategies 
One concern with dichotomizing the blood pressure variable and doing inference is that you've assumed that all patients with blood pressure below 150 mmHg have the same risk due to their blood pressure, which I don't believe is true. I don't think there is any biology behind the assumption that once a patient's blood pressure rises above 150mmHg they magically become at higher risk of a heart attack. It's more likely that small increases in blood pressure lead to small increases in risk.
Incorrect assumptions like this can invalidate the inference because the model is no longer correct. Incorrect models will lead to invalid inferences and incorrect p-values, so it's essential we define the model to best fit what is biologically plausible. This means treating blood pressure as continuous instead of dichotomizing.
