Fundamentally the problem is the Esk, because the only ways across it are Auchendinny* (steep), Roslin Glen (steep and narrow), Polton Bank (steep, narrow and many blind corners) and Lasswade (not so steep but long and busy), and the three crossings by Eskbank/Dalkeith (not so steep, a bit busy). And ultimately, you have to cross the city bypass at Lothianburn (ok), Straiton (ok but busy) or Gilmerton Road†. Which route you go for depends whether you're heading to east of Princes St or west.
From the Granites, my route would be to blast along the A7, and turn left at Dobbies and use the shared path as far as GSR. Then you can drop down to Cameron Toll, etc.
The quieter route is Carrington to Bonnyrigg, NCN196/1 to Eskbank Toll and Melville Road to Dobbies and onwards. I use Melville Road (&c.) when trundling home from Dalkeith Country Park on my Elephant Bike.
To be honest, if I were heading for west Edinburgh, I would either stomach the climbing to Penicuik, and go A701, A703, A702, which is mostly flat; or bang it along to Cameron Toll and turn west to Blackford.
* I didn't mention Penicuik, because from Temple you're climbing pretty much all the way.
† Using the Bilston viaduct and crossing under the bypass is also pointless because you've just had to use a load of energy after crossing the Esk anyway.