I've looked at multiple options including combining fields, scaffolding, and back-end unions, but I think your best option will be to pivot your date fields.
In the Data Source window you can Control + Select the date fields and then right click to select "Pivot". This will create one column for your dates and another column to describe those dates. The one major downside is that this will increase the amount of data since every row will now be multiple rows to accommodate for the change in date. For example, if you have three date fields (Created, Converted, Updated), you will now have three rows for each original row (one for each date type). I'm attaching a pivoted workbook (Superstore Sales) as an example.
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Using multiple date dimension.twbx 605.1 KB