Assuming you're using Postgresql with the PostGIS extension (and assuming all your geometries are linestrings, and that they are in WGS84 coordinate system), it's easy to load this into a QGIS layer using the database manager (from the menu: "Database/DB manager") and the PostGIS function ST_GeomFromText(). Connect to your database, open an SQL window, and run something like this:
SELECT myfield1, myfield2, ST_GeomFromText(geowkt, 4326) AS geom from mytable;
You can then load this as a layer into your QGIS project, selecting geom as the geometry field.
(If you don't have PostGIS installed, you could select the data you want, including your linestring field, and export it as CSV, then import the CSV file into QGIS.)