I would use boxplots to display the relationship between a discrete and a continuous variable. You can make your boxplots vertical or horizontal with standard statistical software, so it's easy to visualize as either IV or DV. It is possible to use a scatterplot with a discrete and continuous variable, just assign a number to the discrete variable (e.g., 1 & 2), and jitter those values (note top plot on right here).
Regarding your comment that the line of best fit might be biased, it depends on what you have. For instance, if you have a discrete variable with two levels as your IV, and a continuous variable as your DV, you can draw a line through the two means and this will not be biased. (We would typically think of this situation as being appropriate for a t-test, but it is actually a form--i.e., simple case--of regression, see my answer here.) On the other hand, if you have a discrete variable with two levels as your DV, standard (OLS) regression would be inappropriate (logistic regression would be called for) and the line of best fit would be biased, but you could fit (& plot) a lowess line as part of your initial data exploration.
For visualizing the relationship between two discrete variables, I would use a mosaic plot. You could also use a sieve plot, an association plot, or a dynamic pressure plot with some programming.