I have a process that I have used for years where I create .ascii files with a Python process, import them to GRASS GIS 6.4.3 and do some raster calculations (r.mapcalc). This works great but then for mapping in QGIS, I have to go back to version 1.8.0 (I'm glad I have saved this old version!) to create output maps of my calculated raster maps. Am I missing something or has QGIS left compatibility with GRASS rasters behind?
You should not have to revert to an old version of QGIS. It sounds like you have not activated GRASS. Have a look at the QGIS documentation. Ensure that you have GRASS installed and can see the tools in the Processing Toolbox.
Next ensure that GRASS is activated as a data provider. In the Processing Toolbox, click on the little spanner icon and expand the Providers section (or go Settings->Processing->Providers). If you don't see GRASS in there, then either you didn't include it in your install (depends on what install method you used but I think it is now in by default) or your paths are wrong (in which case you ought to be getting a warning on startup).
If GRASS is correctly installed and activated then you should be able to see little GRASS icons in the browser window and open locations as per the documentation linked above.
As J.R says - you don't need to flip in and out of GRASS and QGIS. You can use all the great power of GRASS modules directly within QGIS on any raster or vector data without having to go through the tedious process of setting up GRASS locations etc. (so you can use GRASS modules on, say, a TIFF without importing it into GRASS first). This should greatly simplify your workflow - especially as you could create repeatable workflows through the model builder.