The Problem

It can be useful to set-up your data to perform your comparisons with multiple measures, and, for business reasons, a pie visualization is preferred. This article is for who wants different measures to be the values on the pie chart.
You may also be able to resolve this via PDT, but that would require using aggregates of measures as the values of a single dimension.

Note: This works best for relatively small datasets.

For a displaying multiple pie charts (one for each measure) comparing dimension values, see this article: How to display two measure in pie chart.

The Workaround

Set-up your data so that you have a dimension with at least as many unique values as measures you are trying to display.

Add a table calculation to return the intended aggregation of each measure based on rows using nested if statements.
ex:
Add a second table calculation to return the desired value labels for the measure based on row (corresponding to the order used for "Desired Values by row")
ex:
Open the visualization tab if it is not already enabled.
Hide all dimensions/views from the visualization. Only have the table calculations set to display.
Select the pie visualization.