This is a fix that I have pieced together from other posts. It works every time:

if you change either the module or the tab (page) after the PubDate of your Announcement-Items, all your Announcement-Items will get the same PubDate of the last change of the module or tab.

My workaround for this was to manually set the LastChangedOnDate for my announcement modules to an older date then my Announcement-Items PubDates. Therefore I updated the column LastModifiedOnDate in the tables Modules, TabModules and Tabs for all rows with my announcement module ID's. Then I restarted my portal (to get clean caches), after that I cleared the SearchItem* tables and started a reindex (see below on how to re-index the SearchItem* tables).

After applying this workaround the RSS PubDates were in fact the PubDates of the Announcement-Items and stayed stable on further indexer runs.

Hope is of help to anybody.

Best regards,
Theo
.....
SearchItem* tables re-indexing:
If by accident the search index got corrupted, there will be no search results displayed any further and you have to delete the search index tables manually. Follow these steps:
 Login as Superuser ("host" by default)
 In Host Menu select item "SQL"
 Copy the following 4 lines and paste them into the text box:
truncate table {databaseOwner}{objectQualifier}SearchItemWordPosition
DELETE {databaseOwner}{objectQualifier}SearchItemWord
DELETE {databaseOwner}{objectQualifier}SearchWord
DELETE {databaseOwner}{objectQualifier}SearchItem
 Activate the script check box and hit Run
 The search index will be rebuilt automatically by the scheduler.
 Force it with Host, Schedule: edit Search Engine Scheduler; Run Now.

Then on any RSS modules taking this feed, refresh RSS feed cache if necessary.