Yes, you can import aliases for specific fields from dimension tables instead of using joins in a multi-table connection. The easiest way to get started is to create separate data connections for each individual table, and use the Tableau "Data Blending" feature to bring dimension values into your visualizations.
You can also use this approach if you wish to capture the aliases as a one-time operation from a dimension table into the Tableau data model for your fact table. The process is similar to the one described in the following Knowledge Base article, except you will choose the menu item "Edit Primary Aliases..." instead of "Create Primary Group...": http://kb.tableausoftware.com/articles/knowledgebase/how-create-primary-groups-secondary-source
There is some nuance to this process so you may need to experiment and read the product documentation, however I hope the information I provided helps you get started.