I recently downloaded the SSURGO data for Ohio. It came as compressed zip files by county. Within each county zip archive were tabular and spatial data. The tabular data is in the form of CSV text files, and the spatial data as shapefiles. I assume you will start with a similar download.
The NRCS SSURGO website has metadata on the datasets. The SSURGO Metadata - Tables and Columns Report contains information on each of the tables. The text file names are sometimes a bit different than the table names in order to keep text file names at 8 characters, but aren't too hard to correlate. Because the text files do not have header rows, the tables and columns report will tell you the order of data in each row (the columns in the report are listed in the same order as the fields in the text file).
The data model diagrams on the SSURGO website and the relationship report will tell you how all of the tables relate to each other. In the upper right corner of diagram 1 you will see some tables in blue that match the shapefiles downloaded. You can use the indicated fields and relationships to pull the data you need.
Since you are discussing PostGIS and trying to avoid Microsoft, I am assuming you do not have Esri products. If you haven't already, I suggest you look into QGIS. By importing the shapefiles and tables (as CSV) both into QGIS, you can create joins under the layer properties to access the information you want from the SSURGO tables. This way, you avoid MS Access as you desire, and can start to work with the data a bit before going into PostGIS. This may also be a good way to process the data before import to Postgres/PostGIS.
When you are ready to put all this into PostGIS, I suggest you look at this post. There is some good information on it. I've been working on a state-wide download of SSURGO data that includes all of the auxiliary tables. I hope to add links to some of the SQL and scripts I've used in the future, but they aren't quite clean enough to share yet.
FYI - the primary advantage to using the MS Access templates are that you don't need to figure out the field order yourself, and there are some queries already created that relate information between the various tables. You could then connect to the database the same way I discussed connecting to CSV text files, and create a join between the shapefiles and database tables (vs. shapefiles and text files).